Prof Abed-Meraim Karim

Prof Adel Hafiane

Prof Prisme

Linh Nguyen

Trung

Guanglie Ouyang

Zuokun Ouyang

Pham Minh Tuan

Dr Hoang Vy

Thuy Lynh

Vu Thi

An Overview of Tensor Tracking Robust Tensor Tracking with Missing Data and Outliers Tensor Tracking under Tensor-Train Format

Keywords: . . . . . . . . . . . . . tensor subspace learning. . . . . . . . . . . . . . . . . . streaming tensor-train decomposition. . . . . . . . . . . . . . . . . tensors. . . . . . . . . . . . . . . . . . . . . Robust Subspace Tracking with Missing Data and Outliers Sparse Subspace Tracking In High Dimensions Data stream, low-rank approximation, subspace, tensor

This thesis marks the end of my three-year journey towards a Ph.D

List of Figures

3.

12 Effect of outlier intensity on robust matrix completion performance. White color denotes perfect recovery, black color denotes failure and gray colour is in between.

3.13 Qualitative illustration of video background-foreground separation application. .

Notations

R (resp. C) set of real (resp. complex) numbers

x, x, X, X , and X scalar, vector, matrix, tensor, and set/subset/support

x i,j /X(i, j)/[X] ij (i, j)-th entry of X

x i 1 ,...,i N /X(i 1 , . . . , i N )/[X ] i 1 ...i N (i 1 , . . . , i N )-th entry of X

x = vec(X) vectorization of X X = diag(x) diagonal matrix X with x on the main diagonal X(i, :), X(:, j) i-th row and j-th column of X X ⊤ , X -1 X # transpose, inverse, and pseudo-inverse of X λ max (X), λ min (X) largest and smallest singular values of X κ(X) condition number of X equal to λmax(X) λ min (X) rank(X) rank of X span(X) the column space of a tall matrix X tr(X) trace of X θ(X, Y) canonical angle between span(X) and span(Y)

I n n × n identity matrix U (n)
n-th loading factor/matrix X (n) , unfold n (X ) mode-n unfolding of X Y = bcirc(X ) block circulant tensor Y specified by X

•, ⊛, ⊙, ⊗ outer, Hadamard, Khatri-Rao, and Kronecker product

N n=1 U (n) U (N ) ⊙ U (N -1) ⊙ • • • ⊙ U (1) N n=1 U (n) U (N ) ⊗ U (N -1) ⊗ • • • ⊗ U (1)
X × n U n-mode product of X with U,

X × 1 n Y mode-(n, 1) contracted product of X with Y X ⊞ n Y concatenation of X with Y along the n-th mode xvi Notations X * Y t-product of X with Y X ⊆ Y X is a sub-tensor of Y {U (n) } N n=1 r i=1 U (1) (:, i) • U (2) (:, i) • • • • • U (N ) (:, i) X ; {U (n) } N n=1 X × 1 U (1) × 2 U (2) × 3 • • • × N U (N ) ∥.∥ F ,

Big Data Stream Processing

Stream processing has recently attracted much attention from both academia and industry due to the fact that massive data streams have been increasingly collected over the years and they can be smartly mined to discover new insights and valuable information [START_REF] Chen | Data-intensive applications, challenges, techniques and technologies: A survey on Big Data[END_REF][2][START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF]. For example, we are living in the Internet of Things (IoT) era where a huge number of sensing devices have been installed and developed, see Fig. 1.1. These devices have the capability to collect, manage, and transmit data via IoT networks in real time. Accordingly, stream processing is required to retrieve important insights from such IoT data in seconds or even faster for facilitating real-time decision making [4].

In many modern online applications, data streams have three "V"-characteristics: Volume, Velocity, and Veracity. As they are continuously generated, their volume grows significantly over time and possibly to infinity. Thus, one of the most notable features of streaming data is that they are unbounded sequences of data samples. Velocity refers to the high-speed data arrival rate and real-time processing. Data collected from user interactions in social networks (e.g., Facebook, Instagram, and Twitter) are, for example, at very high velocity. Veracity implies the 1.1. BIG DATA STREAM PROCESSING relates to the biasedness, noise, uncertainty, incompleteness, and abnormality in data. Apart from the three "V"s, streaming data have some other distinctive characteristics, including time sensitivity/variation (aka concept drift), heterogeneity (different sources with diversity of data types), volatile and unrepeatable property, and so on [2,[START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF]5,6]. These characteristics lead to several inherent requirements and computational issues for stream processing, such as:

Internet of Things

• Low latency: Stream methods and systems need to efficiently acquire, manage, and process flows of data without introducing additional delays.

• Low memory storage: Stream methods and systems must have the ability to operate in an online fashion with limited memory resources.

• Scalability: As streaming data normally grow in size much faster than computational resources, stream processing requires scalable methods and systems.

• Time variation: As streaming data can evolve with time, stream methods and systems are required to be capable of tracking their variation along the time.

• Robustness: In many cases, streaming data are imperfect and unreliable, so stream methods and systems should have the potential to estimate and compute answers from corrupted observations.

They are, however, also potential benefits of stream processing against batch processing, we refer the readers to Table . 1.1 for a brief comparison between the two kinds of processing.

In this work, we mainly focus on stream methods which are capable of tracking the low-rank approximation (LRA) of big data streams over time. Technically, the primary objective of the LRA is to approximate high-dimensional data by a more compact low-dimensional representation with limited loss of information [7]. Therefore, finding the LRA is a fundamental and essential task for data mining in general and streaming data analytics in particular. For the sake of convenience and convention, in what follows, we first list some linear and multilinear algebraic operations (for vectors, matrices, and tensors) that are frequently used throughout this manuscript. Next, we introduce one of the most well-known linear algebra techniques for finding the LRA of matrices in batch setting, singular value decomposition (SVD), and then describe its connection to some common types of tensor decomposition (TD). Finally, we present their online (adaptive) variants for dealing with streaming data derived from one-dimensional observations (i.e., SVD → subspace tracking) and multi-dimensional observations (i.e., tensor decomposition → tensor tracking).

Vector, Matrix, and Tensor Operations

In this thesis, we use the following notational conventions. Lowercase, boldface lowercase, and boldface capital letters denote scalars (e.g., x), vectors (e.g., x), and matrices (e.g., X), respectively. Calligraphic and bold calligraphic letters are used to represent sets/subsects/supports (e.g., X ) and tensors (e.g., X ), respectively. For index notations, we use x i or x(i) to denote the i-th element of x. The (i, j)-th element, the i-th row, and the j-th column of X are denoted by x i,j or X(i, j), X i,: or X(i, :), and X :,j or X(:, j), respectively. We denote by X ⊤ , X -1 , and X # the transpose, inverse, and pseudo-inverse of X, respectively. The (i 1 , i 2 , . . . , i N )-th element of X is represented by x i 1 ,i 2 ,...,i N , X (i 1 , i 2 , . . . , i N ), or [X ] i 1 ,i 2 ,...,i N . In addition, X :,...,:,in,:,...,: or X (:, . . . , :, i n , :, . . . , :) represents a sub-tensor of X obtained by holding the n-th index of X at i n . The mode-n matricization of X is denoted by X (n) . Symbols ∥.∥ p and ∥.∥ F represent the ℓ p 1.1. BIG DATA STREAM PROCESSING norm and Frobenius norm. In the following, we summarize some useful linear and multilinear algebraic operations, to be used later.

Outer product: Given two vectors x ∈ R N ×1 and y ∈ R M ×1 , their outer product is defined as

follows x • y =         
x 1 y 1 x 1 y 2 . . . x 1 y M x 2 y 1 x 2 y 2 . . . x 2 y M . . . . . . . . . . . .

x N y 1 x N y 2 . . . x N y M          = y 1 x y 2 x . . . y M x ∈ R N ×M . (1.1)
For a generalized case, the outer product of two tensors

X ∈ R I 1 ×I 2 ו••×I N and Y ∈ R J 1 ×J 2 ו••×J M yields a tensor Z = X • Y ∈ R I 1 ו••×I N ×J 1 ו••×J M with elements Z(i 1 , i 2 , .
. . , i N , j 1 , j 2 , . . . , j M ) = X (i 1 , i 2 , . . . , i N )Y(j 1 , j 2 , . . . , j M ).

(1.2)

Kronecker product: Given two matrices X ∈ R N ×M and Y ∈ R P ×Q , the Kronecker product of X and Y results in an N P × M Q matrix of the following form and Y ∈ R M ×r , their Khatri-Rao product is an N M × r matrix of the following form X ⊙ Y = X(:, 1) ⊗ Y(:, 1) X(:, 2) ⊗ Y(:, 2) . . . X(:, r) ⊗ Y(:, r) ∈ R N M ×r .

X ⊗ Y =          x 1,1 Y x 1,2 Y . . . x 1,M Y x 2,
(1.4)

For short, we denote the Kronecker product and Khatri-Rao product of a sequence of matrices (1) .

{U (n) } N n=1 as follows N n=1 U (n) = U (N ) ⊗ U (N -1) ⊗ • • • ⊗ U (1) , (1.5)
N n=1 U (n) = U (N ) ⊙ U (N -1) ⊙ • • • ⊙ U
(1.6)

Tensor unfold and fold operations: The unfold of X ∈ R I 1 ×I 2 ו••×I N , written as unfold(X ),

1.1. BIG DATA STREAM PROCESSING returns a tensor Z of lower order:

Z = unfold X =         
X :,...,:,1 X :,...,:,2 . . . X :,...,:,

I N          ∈ R I 1 I N ×I 2 ×I 3 ו••×I N -1 . (1.7)
Its inverse operator denoted by fold(Z) reshapes Z back to X as fold unfold(X ) = X .

Tensor concatenation: The concatenation of two tensors

X ∈ R I 1 ×I 2 ו••×I N and Y ∈ R I 1 ו••×I N -1 ×W
along the last dimension results in Z = X ⊞ Y ∈ R I 1 ו••×I N -1 ×(I N +W ) with elements

Z(i 1 , i 2 , . . . , i N ) =    X (i 1 , i 2 , . . . , i N ), if i N ≤ I N , Y(i 1 , i 2 , . . . , i N ), if I N + W ≥ i N > I N .
(1.8)

Mode-n product: The mode-n product of a tensor

X ∈ R I 1 ×I 2 ו••×I N with a matrix U ∈ R J×In returns a tensor Z = X × n U ∈ R I 1 ו••×I n-1 ×J×I n+1 ו••×I N with elements Z(i 1 , . . . , i n-1 , j, i n+1 , . . . , i N ) = In in=1
X (i 1 , . . . , i n-1 , i n , i n+1 , . . . , i N )U(j, i n ).

(1.9)

The mode-n product of X with N matrices {U (n) } N n=1 along all N modes is denoted as

X , U (n) N n=1 = X × 1 U (1) × 2 U (2) × 3 • • • × N U (N ) .
(1.10)

Mode-(N, 1) product (aka tensor-train contraction): The mode-(N, 1) product of

X ∈ R I 1 ×I 2 ו••×I N with Y ∈ R I N ×J 2 ו••×J M , written as X × 1 N Y, results in a tensor Z ∈ R I 1 ו••×J N -1 ×J 2 ו••×J M with elements Z(i 1 , . . . , i N -1 , j 2 , . . . , j M ) = I N i N =1
X (i 1 , . . . , i N -1 , i N )Y(i N , j 2 , . . . , j M ).

(1.11)

T-product: The t-product of X ∈ R I 1 ×I 2 ו••×I N and Y ∈ R I 2 ×J×I 3 ו••×I N ,
U I N U I N -1 . . . U 2 U 1          , (1.13)
where U i = U :,...,:,i and the base case of the t-product of two 3-order tensors A ∈ R J 1 ×J 2 ×J 3 and B ∈ R J 2 ×K×J 3 is defined as

A * B = fold bcirc(A) • unfold(B) ∈ R J 1 ×K×J 3 . (1.14)
Inner product: Given two tensors X and Y of the same size

I 1 × I 2 × • • • × I N , their inner
product is defined as

X , Y = I 1 i 1 I 2 i 2 • • • I N i N X (i 1 , i 2 . . . , i N )Y(i 1 , i 2 . . . , i N ). (1.15)
1.1.2 Batch Low-rank Approximation: From SVD to Tensor Decomposition

It is very well known that SVD is one of the most powerful and widely-used linear algebra techniques with a number of applications in various domains [START_REF] Stewart | On the early history of the singular value decomposition[END_REF][START_REF] Golub | Matrix Computations[END_REF]. Particularly, the compact SVD of a rank-r matrix X ∈ R I 1 ×I 2 is given by

X SVD = u 1 , u 2 , . . . , u r U          λ 1 λ 2 . . . λ r          Λ          v ⊤ 1 v ⊤ 2 . . . v ⊤ r          V ⊤ = r i=1 λ i u i v ⊤ i , (1.16)
where U ∈ R I 1 ×r and V ∈ R I 2 ×r are unitary matrices; and Λ ∈ R r×r is a diagonal matrix whose diagonal values are positive, i.e., λ 1 ≥ λ 2 ≥ • • • ≥ λ r > 0, see Fig. 1.2 for an illustration. For the problem of low-rank approximation in batch setting, the following theorem indicates that SVD can give the best LRA for any matrix X. Theorem 1 (Eckart-Young-Mirsky Theorem [START_REF] Golub | Matrix Computations[END_REF]). Denote by X = UΛV ⊤ the SVD of

X ∈ R I 1 ×I 2 . If k ≤ rank(X) and X k = k i=1 λ i u i v ⊤ i , then min
A∈R I 1 ×I 2 rank(A)≤k X -A = X -X k , (1.17)
with respect to both the spectral norm and Frobenius norm.

Thanks to Theorem 1, the best rank-k approximation of X can be obtained by applying the following procedure:

• Step 1 : Compute X SVD = UΛV ⊤ , where U ∈ R I 1 ×I 1 and V ∈ R I 2 ×I 2 are unitary matrices, and the diagonal matrix Λ ∈ R I 1 ×I 2 contains positive diagonal entries in descending order.

• Step 2 : Select the first k singular vectors from U and V to form the following matrices U k = U(:, 1 : k) and V k = V(:, 1 : k).

• Step 3 : Select the top k strongest singular values in Λ to form: Λ k = Λ(1 : k, 1 : k).

• Step 4 : Derive the best rank-k approximation of X from:

X k = U k Λ k V ⊤ k .
When dealing with tensors (aka, multidimensional arrays), several multiway extensions of the SVD have been developed for tensor decomposition (TD) in the literature [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF][START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions[END_REF][START_REF] Liu | Tensor Computation for Data Analysis[END_REF]. The five common types of TD are CP/PARFAC [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF], Tucker/HOSVD [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF], tensor-train/network [START_REF] Oseledets | Tensor-train decomposition[END_REF],

t-SVD [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF], and block-term decomposition (BTD) [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF], see Fig. 1.3 for illustrations. Specifically, they aim to factorize a tensor into a set of basis components (e.g., vectors, matrices, or simpler tensors) and hence offer good low-rank tensor approximations. In the following, we describe their connection to SVD and refer the readers to Chapter 5 for further details on their main features, properties, and algorithms.

CP/PARAFAC Decomposition: Similar to SVD that represents X by a sum of rank-1 matrices (i.e., λ i u i v ⊤ i ), the CP decomposition also factorizes a tensor X ∈ R I 1 ×I 2 ו••×I N into rank-1 terms:

X CP = r i=1 λ i u (1) i • u (2) i • • • • • u (N ) i rank-1 term , (1.18)
1.1. BIG DATA STREAM PROCESSING

(: ) ,r U [START_REF] Chen | Data-intensive applications, challenges, techniques and technologies: A survey on Big Data[END_REF] (: ∈ R In×1 with 1 ≤ n ≤ N plays the same role as singular vectors of U and V in the SVD model (1.16) (note that u i v ⊤ i = u i • v i ) [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF]. The matrix

) ,r U      (2) 1 U 1  2  N  1 G (1) 1 U ( ) 1 N U     V G  1 1  1 2  1 N  (1) G (2) G ( 1) N  G ( ) N G 1 1 N    (1)
U (n) = u (n) 1 , u (n)
2 , . . . , u

(n) r is the n-th CP factor of X and it is not required to be orthogonal. Following the general definition of matrix rank, the smallest integer r satisfying (1.18) is referred to as the tensor (CP) rank of X . Under certain conditions, CP decomposition is essentially unique up to a permutation and scale which is an useful property in many applications.

Tucker/HOSVD Decomposition: Apart from the classical form (1.16), we can express the SVD of X as follows

X SVD = UΛV ⊤ = Λ core × 1 U × 2 V 2 factors
.

(1. [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF] Accordingly, a direct multiway extension of (1.19) to high-order tensors can be given by

X Tucker = G core × 1 U (1) × 2 U (2) × 3 • • • × N U (N ) N factors , (1.20)
where the core G ∈ R r 1 ×r 2 ו••×r N is a tensor of smaller size than X (i.e., r n ≤ I n ∀n) and N tensor factors {U (n) } N n=1 , U (n) ∈ R In×rn are orthogonal matrices. The representation model (1.20) is regarded as the high-order SVD (HOSVD) or Tucker format [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF]. Unlike the SVD and CP, 1.1. BIG DATA STREAM PROCESSING Tucker/HOSVD is not unique in general. However, as the subspace covering U (n) is physically unique, its main objective is for finding principal subspaces of the tensor factors [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF].

Block-Term Decomposition: BTD factorizes X into several blocks of low multilinear-rank instead of rank-1 terms

X BTD = r i=1 G i × 1 U (1) i × 2 U (2) i × 3 • • • × N U (N ) i low multilinear-rank term .
(1.21)

The BTD can be viewed as a unification and generalization of the two well-known CP and Tucker decompositions. Specifically, when {G i } r i=1 are diagonal tensors, BTD boils down to the CP decomposition. It has the form of Tucker decomposition when only one block term (i.e., r = 1) is considered. In addition, several appealing features of the BTD are inherited from CP and Tucker such as stable computation of Tucker, identification and uniqueness of CP [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF]. In parallel, it is worth recalling a remark in [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF] that "the rank of a higher-order tensor is actually a combination of the two aspects: one should specify the number of blocks and their size". That means BTD provides a unified approach to generalize the concept of matrix rank to tensors.

Tensor-Train Decomposition: Together with (1. [START_REF] Oseledets | Tensor-train decomposition[END_REF]) and (1.19), we can write the SVD of X as

X(i 1 , i 2 ) SVD = r k=1 λ k U(i 1 , k)V(k, i 2 ). (1.22)
Accordingly, each element of a high-order tensor X can be represented by

X (i 1 , i 2 , . . . , i N ) TT = r 1 ,r 2 ,...,r N -1 k 1 ,k 2 ,...,k N -1 G 1 (1, i 1 , k 1 )G 2 (k 1 , i 2 , k 2 ) . . . G N (k N -1 , i N , 1). (1.23)
where G n is an r n-1 × I n × r n tensor with n = 1, 2, . . . , N -1 and r 0 = r N = 1. We refer to the representation model (1.23) as tensor-train (TT). Like CP, the TT format offers a memory-saving model for representing high-order tensors. Like Tucker, the TT decomposition and the TT rank r = [r 1 , r 2 , . . . , r N -1 ] of any tensor X can be numerically computed in a stable and efficient way.

t-SVD Decomposition: Last but not least, another extension of SVD to high-order tensors is the tensor SVD (t-SVD) which is of the following form:

X t-SVD = U orthogonal * G f -diagonal * V orthogonal , (1.24)
where U and V are unitary tensors, and G is a rectangle f -diagonal tensor whose frontal slices are diagonal matrices [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF]. Intuitively, the t-SVD model (1.24) shares the similar form with the SVD in (1.16). However, due to the t-product " * ", the algebraic framework used in the t-SVD is quite different from the classical (multi)-linear algebra in other types of TD and SVD.

1.1. BIG DATA STREAM PROCESSING For example, most of its computations are performed in the Fourier domain. Under the t-SVD format, the tubal-rank which is equal to the number of non-zero tubes of G is used to define the LRA of tensors in the same manner as the SVD.

1.1.3 Online Low-rank Approximation: From Subspace to Tensor Tracking

In online setting, data samples are continuously collected with time. Accordingly, recomputing the batch LRA methods (e.g., SVD or batch TD algorithms) at each time step becomes inefficient due to their high complexity and the time variation, aka concept/distribution drift. This has led to defining a variant of the LRA called online (adaptive) LRA in which we may want to track the underlying process that generates streaming data with time.

When observations arriving at each time are one-dimensional (i.e., vectors), the main interest in the online LRA is to estimate the principal subspace that compactly spans these observations over time. Specifically, it is referred to as the problem of subspace tracking (ST) in signal processing which has been developed for over three decades [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF][START_REF] Delmas | Subspace tracking for signal processing[END_REF][START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF]. In general, on the arrival of the new data y t ∈ R I 1 ×1 at time t, the subspace matrix U t ∈ R I 1 ×r can be derived from analysing the spectrum of the following covariance matrix

C t = t τ =t-Lt+1 β t-τ y τ y ⊤ τ , (1.25)
where L t is the window length and 0 < β ≤ 1 is the forgetting factor [START_REF] Delmas | Subspace tracking for signal processing[END_REF]. When L t = t and β = 1, C t in (1.25) boils down to the classical sample covariance matrix. More specifically, in a connection to the batch LRA using SVD, the vector y t can be seen as the t-th column of the underlying matrix X t = [X t-1 y t ], see Fig. 1.4 for an illustration. The subspace matrix U t plays a role as the left singular vector matrix of X t , while the coefficient vector w t = U ⊤ t y t is indeed the t-th row of the matrix VΛ in the SVD expression (1.16). Depending on the choice of C t and the subspace estimation technique, we can obtain several subspace tracking algorithms.

When observations arriving at each time are multidimensional (i.e., tensors), the online LRA turns out to be tensor tracking which can be considered as a generalization of subspace tracking.

In particular, we wish to estimate the tensor dictionary (e.g., core tensor(s) and tensor factors) that generates the underlying streaming data X t over time:

X t =    X t-1 ⊞ Y t if single-aspect streaming X t-1 ∪ Y t if multi-aspect streaming , (1.26)
where " ⊞ " and " ∪ " denote the tensor concatenation and union operator, while X t-1 and Y t represent the old and new observations, respectively. The "single-aspect streaming" model and the "multi-aspect streaming" model are, respectively, dedicated to represent data streams In recent years, the explosion of big data streams have posed significant challenges to the online LRA problem. For example, efficiency and robustness are highly important when we deal with streaming data in high dimensions. Many theoretical results in random matrix theory (e.g., [START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF][START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Vershynin | How close is the sample covariance matrix to the actual covariance matrix?[END_REF]) indicated that the sample covariance matrix (SCM) is not an efficient estimator of the actual covariance matrix in the high-dimension, low-sample-size regime where datasets are massive in both dimension and sample size. However, most of the state-of-the-art subspace tracking methods in the literature are mainly based on the spectral analysis of the SCM, and thus, they are not effective in such a regime. In parallel, sparse outliers and missing data become more and more ubiquitous in modern streaming applications [6]. Sparse outliers are data points that appear to be inconsistent with or exhibit abnormal behaviour different from others. Missing data are often encountered during the acquisition and collection. Both sparse outliers and missing data can cause several issues for knowledge discovery from data in general and data streams in particular, see Fig. 1.5 for an illustration of outlier's impact on the standard principal component analysis (PCA) which specifically uses SVD in its computation. Therefore, it requires robust algorithms capable of dealing with such data corruptions with time. In addition, scalable tracking algorithms are always desirable for handling modern data streams, especially dealing with large-scale and high-multidimensional data streams. As indicated later, most of the existing tracking algorithms are of high complexity with respect to both computation and memory storage. Accordingly, it is essential to develop efficient and scalable tracking techniques of low cost. In this work, we aim to develop efficient and effective tracking algorithms which have the capability to deal with such challenges.

1 t X t X Y t

Thesis Description

Thesis Outline and Contributions

The rest of my thesis is organized into two major parts addressing respectively subspace tracking and tensor tracking, followed by the conclusion and outlook, please see Fig. A.5 for an overview.

Part I: Subspace Tracking

In Chapter 2, we provide a brief survey on recent robust subspace tracking algorithms which were mostly developed over the last decade. Particularly, we begin by introducing the basic ideas of the subspace tracking problem. We then highlight main classes of algorithms for dealing with non-Gaussian noises (e.g., sparse outliers, impulsive noise, and colored noise). Recent years have also witnessed the widespread of high-dimensional data analysis in which sparse representationbased methods are successfully applied to many signal processing applications. Accordingly, the state-of-the-art sparse subspace tracking algorithms are also reviewed therein.

In Chapter 3, we propose a novel algorithm, namely PETRELS-ADMM, to deal with subspace tracking in the presence of outliers and missing data. The proposed approach consists of two main stages: outlier rejection and subspace estimation. In the first stage, alternating direction INTRODUCTION method of multipliers (ADMM) is effectively exploited to detect outliers affecting the observed data. In the second stage, we propose an improved version of the parallel estimation and tracking by recursive least squares (PETRELS) algorithm to update the underlying subspace in the missing data context. We then present a theoretical convergence analysis of PETRELS-ADMM which shows that it generates a sequence of subspace solutions converging to the optimum of its batch counterpart. The effectiveness of the proposed algorithm, as compared to state-of-the-art algorithms, is illustrated on both simulated and real data.

In Chapter 4, we develop a new provable effective method called OPIT for tracking the sparse principal subspace of data streams over time. Particularly, OPIT introduces a new adaptive variant of power iteration with space and computational complexity linear to the data dimension.

THESIS DESCRIPTION

In addition, a new column-based thresholding operator is developed to regularize the subspace sparsity. Utilizing both advantages of power iteration and thresholding operation, OPIT is capable of tracking the underlying subspace in both classical regime and high dimensional regime.

We also present a theoretical result on its convergence to verify its consistency in high dimensions.

Several experiments are carried out on both synthetic and real data to demonstrate the tracking ability of OPIT.

Part II: Tensor Tracking

In Chapter 5, we provide a contemporary and comprehensive survey on different types of tensor tracking techniques. We particularly categorize the state-of-the-art methods into three main groups: streaming CP decompositions, streaming Tucker decompositions, and streaming decompositions under other tensor formats (i.e., tensor-train, t-SVD, and BTD). In each group, we further divide the existing algorithms into sub-categories based on their main optimization framework and model architectures. Specifically, four main groups of streaming CP decomposition algorithms were emphasized, including subspace-based, block-coordinate descent, Bayesian inference, and multi-aspect streaming decompositions. We categorized the current streaming Tucker decomposition methods into three major classes based on their model architecture. They are online tensor dictionary learning, tensor subspace tracking, and multi-aspect streaming decompositions. Finally, a brief survey on the existing methods which are capable of tracking tensors under TT, BTD, and t-SVD formats is presented.

In Chapter 6, we propose three novel adaptive algorithms for tracking higher-order streaming tensors with time, including ACP, ATD, and RACP. Under the CP format, ACP minimizes an exponentially weighted recursive least-squares cost function to obtain the tensor factors in an efficient way, thanks to the alternative minimization framework and the randomized sketching technique. Under the Tucker format, ATD first tracks the underlying low-dimensional subspaces covering the tensor factors, and then estimates the core tensor using a stochastic approximation.

Both the two algorithms ACP and ATD are fast and fully capable of tracking streaming tensors from incomplete observations. When observations are corrupted by sparse outliers, we introduce the so-called RACP algorithm robust to gross corruptions. Particularly, RACP first performs online outlier rejection to accurately detect and remove sparse outliers, and then performs tensor factor tracking to efficiently update the tensor factors. Convergence analysis for three algorithms are established in the sense that the sequence of generated solutions converges asymptotically to a stationary point of the objective function. Extensive experiments are conducted on both synthetic and real data to demonstrate the effectiveness of the proposed algorithms in comparison with state-of-the-art adaptive algorithms.

In Chapter 7, we introduce three new methods for the problem of streaming tensor-train decomposition. The first method called TT-FOA is capable of tracking the low-rank components 1.2. THESIS DESCRIPTION of high-order tensors from noisy and high-dimensional data with high accuracy, even when they come from time-dependent observations. The second method called ATT is particularly designed for handling incomplete streaming tensors. ATT is scalable, effective, and adept at estimating low TT-rank component of streaming tensors. Besides, ATT can support parallel and distributed computing. To deal with sparse outliers, we propose the so-called ROBOT which stands for ROBust Online Tensor-Train decomposition. Technically, ROBOT has the ability to tracking streaming tensors from imperfect streams (i.e., due to noise, outliers, and missing data) as well as tracking their time variation in dynamic environments.

Conclusion and Outlook

Chapter 8 concludes the thesis with our main results and an outlook to future works. Particularly, we present several research challenges and open problems that should be considered for the development of tracking the low-rank component of data streams in the future. They are data imperfection and corruption; rank revealing and tracking; efficient and scalable tensor tracking; and other aspects such as theoretical analysis, symbolic data, and tracking under some less common tensor formats. Possible solutions for these challenges are also discussed.

List of Publications

Most of the above results have been published/submitted in the following papers:

Journal Papers: Particularly in [37,38], we have addressed the problem of analyzing the theoretical performance limit of system identification techniques under the misspecification of the channel order through the lens of the misspecified Cramer-Rao bound (MCRB) -which is an extension of the well-known Cramer-Rao bound (CRB) when the underlying system model is misspecified. Specifically, we have introduced a new interpretation of the MCRB, called the generalized MCRB (GMCRB), via the Moore-Penrose inverse operator. This bound is useful for singular problems and particularly blind channel estimation problems in which the Hessian matrix is noninvertible.

Two closed-form expressions of the GMCRB are derived for unbiased blind estimators when the channel order is misspecified. The first bound deals with deterministic models where both the 1.2. THESIS DESCRIPTION channel and unknown symbols are deterministic. The second one is devoted to stochastic models where we assume that transmitted symbols are unknown random variables i.i.d. drawn from a Gaussian distribution. Two case studies of channel order misspecification are investigated to demonstrate the effectiveness of the proposed GMCRBs over the classical CRBs. In [39], we have investigated the effect of different prior about communications channels (e.g., specular channel model, finite memory linear time invariant channel model, misspecification caused by array calibration errors, so on) on the performance of semi-blind channel identification for MIMO-OFDM systems.

Part I

Principal component analysis (PCA) and subspace estimation (SE) are popular data analysis tools and

used in a wide range of applications. The main interest in PCA/SE is for dimensionality reduction and low-rank approximation purposes. The emergence of big data streams have led to several essential issues for performing PCA/SE. Among them are (i) the size of such data streams increases over time, (ii) the underlying models may be time-dependent, and (iii) problem of dealing with the uncertainty and incompleteness in data. A robust variant of PCA/SE for such data streams, namely robust online PCA or robust subspace tracking (RST), has been introduced as a good alternative. The main goal of this chapter is to provide a brief survey on recent RST algorithms in signal processing. Particularly, we begin this survey by introducing the basic ideas of the RST problem. Then, different aspects of RST are reviewed with respect to different kinds of non-Gaussian noises and sparse constraints. Our own contributions on this topic are also highlighted.

Introduction

Principal component analysis (PCA) and subspace estimation (SE) are widely used as a fundamental step for dimensionality reduction and analysis. Their main purpose is to extract low-dimensional subspaces from high-dimensional data while still keeping as much relevant information as possible. Consequently, PCA and SE have found success in a wide range of fields, from finance to neuroscience, with the most successful applications in computer science. The main difference between them is that PCA emphasizes the use of eigenvectors rather than of subspace as in SE. PCA in a standard set-up can be implemented by using either eigenvalue decomposition (EVD) or singular value decomposition (SVD) and is proved to be optimal in terms of the Frobenius-norm approximation error by the Eckart-Young theorem [START_REF] Jolliffe | Principal Component Analysis[END_REF].

Recent years have witnessed an increasing interest in adaptive processing [2]. It is mainly due to the fact that online applications generate a huge amount of data streams over time and such streams are often with high veracity and velocity. It is known that veracity requires robust algorithms for handling imperfect data while velocity demands (near) real-time processing.

Accordingly, important classes of PCA, such as subspace tracking (ST) also called PCA for streaming data or streaming PCA or dynamic PCA, and ST with missing data have drawn much research attention recently in signal processing and modern data analysis.

The attractive point of ST resides on two aspects. First, in a similar manner to batch subspace methods [START_REF] Delmas | Subspace tracking for signal processing[END_REF], both the main components and the disturbance components of data observation can be exploited in many different ways. In fact, the subspace is simple to understand (i.e., in a statistical sense) and implement, thus proving its efficiency in many practical applications.

Second, different from batch subspace methods, ST has a better trade-off between the accuracy and the computational complexity, thus making it suitable for time-sensitivity and real-time 2.1. INTRODUCTION applications. Due to its practical use, we can find a wide range of applications in diverse fields [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF][START_REF] Delmas | Subspace tracking for signal processing[END_REF][START_REF] Balzano | Streaming PCA and Subspace Tracking: The Missing Data Case[END_REF], for example, direction of arrival (DoA) tracking in radar and sonar, data compression and filtering, blind channel estimation and equalization, and pattern recognition, to name a few. However, it is well-known that PCA/SE is very sensitive to data corruptions. This fact remains across the above important PCA classes in general and ST in particular. PCA dealing with impulsive noise and outliers is referred to as robust PCA. In 2011, it was revisited in a seminal work of Candes et al [START_REF] Candès | Robust principal component analysis?[END_REF]. This work has attracted many research studies and applications, with over 4000 citations as of now. PCA for streaming data with impulsive noise and outliers is referred to as robust subspace tracking (RST). It is considered much more difficult than the original ST [START_REF] Vaswani | Rethinking PCA for modern data sets: Theory, algorithms, and applications[END_REF].

ST algorithms have been developed for over three decades [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF][START_REF] Delmas | Subspace tracking for signal processing[END_REF]. It has been around ten years since Delma's survey [START_REF] Delmas | Subspace tracking for signal processing[END_REF] and we thus believe it is not only important but the right time to do an up-to-date survey in order to highlight some aspects that were not mentioned in [START_REF] Delmas | Subspace tracking for signal processing[END_REF] as well as recent advances on this topic.

Related Work

Due to the importance of ST, there have been a number of published surveys in the literature.

One of the first and earliest surveys on principal subspace tracking algorithms was carried on by Comon and Golub in [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF]. The survey focuses on methods with high and moderate computational complexity for tracking the low-rank approximation of covariance matrices which may be slowly varying with time. In [START_REF] Delmas | Subspace tracking for signal processing[END_REF], Delmas provided a comprehensive overview on developments of classical ST algorithms with low (linear) complexity.

Recently, different adaptations of PCA for modern datasets and applications were reviewed in [START_REF] Jolliffe | Principal component analysis: A review and recent developments[END_REF]. However, PCA for streaming data or ST was not addressed. The problem of tracking the underlying subspace of data from incomplete observations was discussed in [START_REF] Balzano | Streaming PCA and Subspace Tracking: The Missing Data Case[END_REF] and [START_REF] Wang | Subspace estimation from incomplete observations: A high-dimensional analysis[END_REF].

Particularly, the former concerned methodological classes of ST algorithms that are able to deal with missing data while the latter presented a high-dimensional framework for analyzing their convergence behavior. The survey in [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF] carried out reviews on robust PCA, RST, and robust subspace recovery in the presence of sparse outliers. Two similar surveys to [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF] have also been conducted in [START_REF] Vaswani | Static and dynamic robust PCA and matrix completion: A review[END_REF] and [START_REF] Lerman | An overview of robust subspace recovery[END_REF] which respectively review (i) static and dynamic RPCA algorithms, and (ii) the entire body of works on robust sparse recovery. In the literature, there exist two others surveys on two adaptations of PCA which are distributed PCA [START_REF] Wu | A review of distributed algorithms for principal component analysis[END_REF] and sparse PCA [START_REF] Zou | A selective overview of sparse principal component analysis[END_REF].

The main contributions of the above-mentioned papers are summarized in Table 2.1. [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF]1990] Principal ST A survey on numerical methods for tracking the low-rank approximation of covariance matrices slowly varying with time. [START_REF] Delmas | Subspace tracking for signal processing[END_REF]2010] Principal and minor ST A comprehensive survey on classical ST algorithms. [START_REF] Jolliffe | Principal component analysis: A review and recent developments[END_REF]2016] Principal component analysis A survey on adaptations of PCA for modern datasets and applications.

INTRODUCTION

[ [START_REF] Wang | Subspace estimation from incomplete observations: A high-dimensional analysis[END_REF]2018] Principal ST A high-dimensional analysis framework for the state-of-the-art ST algorithms from incomplete observations. [START_REF] Balzano | Streaming PCA and Subspace Tracking: The Missing Data Case[END_REF]2018] ST and streaming PCA A survey on both classical and recent ST algorithms able to handle missing data and their performance guarantee. [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF]2018] Robust subspace learning A survey on robust PCA, RST, and robust subspace recovery in the presence of sparse outliers. [START_REF] Vaswani | Static and dynamic robust PCA and matrix completion: A review[END_REF]2018] Robust PCA A survey on statistic and dynamic robust PCA algorithms. [START_REF] Wu | A review of distributed algorithms for principal component analysis[END_REF]2018] Principal component analysis A survey on distributed PCA algorithms. [START_REF] Lerman | An overview of robust subspace recovery[END_REF]2018] Robust subspace recovery A survey on works on robust subspace recovery when measurements are corrupted by sparse outliers. [START_REF] Zou | A selective overview of sparse principal component analysis[END_REF]2018] Sparse PCA A survey on recent theoretical developments of sparse PCA.

Ours RST A survey on RST algorithms in the presence of different kinds of corruptions (e.g. outliers, missing data, impulsive, and colored noise) and sparse subspace.

ROBUST SUBSPACE TRACKING: PROBLEM FORMULATION

Main Contributions

To the best of our knowledge, we are not aware of any work that reviews the RST problem in the presence of different kinds of non-Gaussian noise. Although the three surveys [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF][START_REF] Vaswani | Static and dynamic robust PCA and matrix completion: A review[END_REF][START_REF] Lerman | An overview of robust subspace recovery[END_REF] reviewed some classes of RST algorithms, they only discussed on sparse outliers. Methods for other non-Gaussian noises (e.g., impulsive noise and colored noise) have not been reviewed yet. Moreover, no survey exists on the problem of sparse ST in the literature. This observation motivates us to carry out a survey on the topic.

The main goal of this survey is to fill the gap in the literature addressing the following three

Robust Subspace Tracking: Problem Formulation

At each time t, we suppose to observe a signal x t ∈ R n satisfying

x t = P t (ℓ t + v t ), (2.1)
where P t ∈ R n×n is an observation mask matrix indicating the i-th entry of x t is observed (i.e., P t (i, i) = 1) or not (i.e., P t (i, i) = 0), v t ∈ R n×1 is the (non-Gaussian) noise vector and ℓ t is the true signal living in a fixed or slowly time-varying low-dimensional subspace of R n . More concretely, ℓ t = U t w t in which w t is a weight vector and we wish to estimate a rank-r matrix U t such that it can cover the span of the complete-data noiseless signal ℓ t .

U t ∈ R n×r (r ≪ n) is a basis matrix with d(U t , U t-1 ) ∆ = sin θ(U t , U t-1 ) ≪
In this chapter, we consider the RST problem in the presence of different kinds of the non-Gaussian noise v t : sparse outliers, impulse noise, and colored noise. Also, we review sparse ST algorithms under the constraint that the basis matrix U t is sparse.

ROBUST SUBSPACE TRACKING IN THE PRESENCE OF MISSING DATA AND OUTLIERS

Robust Subspace Tracking in the Presence of Missing Data and Outliers

In the literature, there have been several studies on ST in the presence of outliers and missing data. The proposed RST algorithms can be categorized into four main classes: (i) Grassmannian, (ii) recursive east-Squares (RLS), (iii) recursive projected compressive sensing (ReProCS), and

(iv) adaptive projected subgradient method (APSM). We summarize all the RST algorithms robust to outliers and missing data in Table 2.2.

Grassmannian Algorithms

Many of RST algorithms are based on the Grassmannian approach in which the ST procedure can be cast into an optimization process on a Grassmann manifold. More concretely, Grassman manifold is a space that parameterizes all r-dimensional linear subspaces of the N -dimensional vector space. The underlying subspace can be derived from averaging the column span of the (fully or partially) observed signals on the Grassmannian. Interestingly, each observed signal ℓ t spans a one-dimensional subspace which can be described as a point in the Grassmannian.

Therefore, the Grassmannian approach offers several advantages such as a lower number of parameters to optimize and limited memory usage and the resulting RST algorithms are often efficient and scalable to high dimensional data [START_REF] Hauberg | Scalable robust principal component analysis using Grassmann averages[END_REF].

State-of-the-art RST algorithms include GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], GOSUS [START_REF] Xu | GOSUS: Grassmannian online subspace updates with structured-sparsity[END_REF], pROST [START_REF] Seidel | pROST: A smoothed lp-Norm robust online subspace tracking method for background subtraction in video[END_REF][START_REF] Hage | Robust PCA and subspace tracking from incomplete observations using ℓ 0 -Surrogates[END_REF], and RoIGA [START_REF] Chakraborty | Intrinsic Grassmann averages for online linear and robust subspace learning[END_REF][START_REF] Chakraborty | Intrinsic Grassmann averages for online linear, robust and nonlinear subspace learning[END_REF]. In [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], He et al. proposed an efficient RST algorithm called Grassmannian robust adaptive ST (GRASTA) which is a robust version of GROUSE in [START_REF] Balzano | Online identification and tracking of subspaces from highly incomplete information[END_REF]. GRASTA first uses an ℓ 1 -norm cost function to reduce the effect of sparse outliers and then performs the incremental gradient on the Grassmann manifold of the subspace U. In [START_REF] Xu | GOSUS: Grassmannian online subspace updates with structured-sparsity[END_REF], Xu et al. introduced an effective algorithm namely GOSUS for tracking subspace with structured-sparsity. GOSUS also incorporates an adaptive step-size for the incremental gradient on the manifold. The effectiveness of GOSUS was demonstrated via the real application of video background subtraction and multiple face tracking. In [START_REF] Seidel | pROST: A smoothed lp-Norm robust online subspace tracking method for background subtraction in video[END_REF][START_REF] Hage | Robust PCA and subspace tracking from incomplete observations using ℓ 0 -Surrogates[END_REF], Hage et al. proposed a method, namely pPOST that combines the advantages of Grassmannian optimization with a non-convex sparsity measure. Instead of using the ℓ 1 -norm regularization, pPOST uses the penalty with non-convex ℓ 0 -surrogates allows reconstruction even in the case when ℓ 1 -based methods fail. Another algorithm dubbed robust intrinsic Grassmann average (RoIGA) was proposed by Rudrasis et al. in [START_REF] Chakraborty | Intrinsic Grassmann averages for online linear and robust subspace learning[END_REF][START_REF] Chakraborty | Intrinsic Grassmann averages for online linear, robust and nonlinear subspace learning[END_REF]. RoIGA is a geometric approach to computing principal linear subspaces in finite and infinite dimensional reproducing kernel Hilbert spaces. Among them, RoIGA is shown as one of the fastest RST algorithms for handling missing data corrupted by outliers. [START_REF] Seidel | pROST: A smoothed lp-Norm robust online subspace tracking method for background subtraction in video[END_REF][START_REF] Hage | Robust PCA and subspace tracking from incomplete observations using ℓ 0 -Surrogates[END_REF]) [START_REF] Chouvardas | An Adaptive Projected Subgradient based algorithm for robust subspace tracking[END_REF][START_REF]Robust subspace tracking with missing entries: The set-theoretic approach[END_REF]) [25,[START_REF] Thanh | Robust subspace tracking with missing data and outliers via ADMM[END_REF]) + RLS

GRASTA ℓ1-norm + ADMM ✓ ✓ ✗ random ✗ O(nr + r 3 ) (2012 [50]) + Grassmannian GOSUS ℓ2-norm + ADMM ✗ ✓ ✓ random ✗ - (2014 [51]) + Grassmannian pROST ℓ0-norm + Grassmannian ✗ ✓ ✗ random ✗ - (2014
+ Conjugate Gradient MRMD Online max-norm ✗ ✓ ✓ random ✓ - (2014 [54]) regularization ROSETA ℓ1,2-norm + ADMM + ✓ ✓ ✗ random ✗ O(nr 2 ) (2015 [55]) RLS Roubst STAPSM APSM + CoSAMP * ✓ ✓ ✗ random ✓ O(knr 2 ) (2015
ReProCS-cPCA ReProCS ✗ ✓ ✓ batch ✓ O(nr log 2 (n) log(1/ϵ)) ⋄ (2016 [58]) OTNNR Truncated nuclear-norm ✗ ✓ ✗ random ✗ - (2016 [59]) regularization OLP-RPCA ℓp-norm + singular ✗ ✓ ✗ random ✓ O(nr + r 3 ) (2017 [60]) value thresholding L1-PCA ℓ1-norm ✗ ✓ ✗ batch ✗ O(nrω 2 ) ‡ (2018 [61]) + Bit-flipping PETRELS-CFAR Robust statistic + RLS ✓ ✓ ✓ batch ✗ O(nr 2 + nω) + (2018 [62]) s-ReProCS ReProCS ✓ ✓ ✓ batch ✓ O(nr log(n) log(1/ϵ)) ⋄ (2019 [63]) NORST-miss ReProCS ✓ ✗ ✓ batch ✓ O(nr log(1/ϵ)) ⋄ (2019 [64]) L1-IRW ℓ1-norm ✗ ✓ ✗ batch ✗ O(k(nwr 3 p + 2 r nr 2 )) † (2019 [65]) + Bit-flipping OSTP Schatten quasi-norm ✗ ✓ ✗ random ✓ O(nr 2 ) (2019 [66]) + Block-proximal gradient NORST ReProCS ✓ ✓ ✓ batch ✓ O(nr log(1/ϵ)) ⋄ (2020 [67]) RoIGA IGA # + Grassmannian ✗ ✓ ✗ random ✗ - (2020 [68, 69]) PETRELS-ADMM ℓ1-norm + ADMM ✓ ✓ ✓ random ✓ O(nr 2 ) (2021

Recursive Least-Squares based Algorithms

Another line of the RST research is based on recursive least-squares (RLS) methods where the underlying subspace is recursively updated by minimizing a (weighted) least-squares objective function containing squared residuals and a penalty accounting for outliers. An efficient RLSbased algorithm is parallel estimation and tracking by recursive least squares (PETRELS) [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF] which can be considered as an extension of the projection approximation ST (PAST) algorithm [START_REF] Yang | Projection approximation subspace tracking[END_REF] in order to handle missing data.

Inspired by PETRELS, several robust variants have been proposed to deal with outliers the same line such as [25,[START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF][START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF][START_REF] Thanh | Robust subspace tracking with missing data and outliers via ADMM[END_REF]. Robust online subspace estimation and tracking (ROSETA) in [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF] applies an adaptive step size at the stage of subspace estimation to enhance the convergence rate. Meanwhile the main idea of PETRELS-CFAR algorithm [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] is to handle "outliers-removed" data (i.e., outliers are first removed before performing ST) using a Constant False Alarm Rate (CFAR) detector. Adopting the approach of PETRELS-CFAR, but aiming to improve RST performance, we proposed an efficient algorithm called PETRELS-ADMM which is able to remove outliers more effectively in [25,[START_REF] Thanh | Robust subspace tracking with missing data and outliers via ADMM[END_REF]. It includes two main stages: outlier rejection and subspace estimation and tracking. Outliers living in the measurement data are detected and removed by a ADMM solver in an effective way. An improved PETRELS was then introduced to update the underlying subspace. In practice, the convergence rate of RST-type algorithms is often faster than that of Grassmmannian-based algorithms in slowly time-varying environments.

Recursive Projected Compressive Sensing based Algorithms

Recursive projected compressive sensing (ReProCS)-based algorithms [START_REF] Zhan | Online (and offline) robust PCA: Novel algorithms and performance guarantees[END_REF][START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF][START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF][START_REF] Narayanamurthy | Fast robust subspace tracking via PCA in sparse data-dependent noise[END_REF] are also capable of tracking subspace in the presence of outliers and missing data.

ReProCS-type algorithms use the piecewise constant subspace change model described previously and start with a "good" estimate of the initial subspace. At each time, they first solve a projected compressive sensing problem to derive the sparse outliers, e.g., using ℓ 1 minimization followed by thresholding-based support estimation. After that, the subspace direction change is then estimated by using projection-SVD [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF].

ReProCS provides not only a memory-efficient and highly robust solution, but also a precise subspace estimation compared to the state-of-the-arts. However, ReProCS-type algorithms often require strong assumptions on subspace changes, outlier magnitudes, and accurate initialization.

Adaptive Projected Subgradient Method based Algorithms

Adaptive projected subgradient method (APSM) can provide a robust solution to the presence of missing data and outliers [START_REF] Chouvardas | An Adaptive Projected Subgradient based algorithm for robust subspace tracking[END_REF][START_REF]Robust subspace tracking with missing entries: The set-theoretic approach[END_REF]. Main advantages of APSM are that convex constraints can be greedy techniques (e.g. compressed sampling orthogonal matching pursuit as used in [START_REF]Robust subspace tracking with missing entries: The set-theoretic approach[END_REF]) and then reject them.

Other Algorithms

Some other RST algorithms are able to track the underlying subspace over time from measurements corrupted by sparse outliers such as MRMD [START_REF] Shen | Online optimization for max-norm regularization[END_REF], OTNNR [START_REF] Hong | Online robust principal component analysis via truncated nuclear norm regularization[END_REF], L1-PCA [START_REF] Markopoulos | Adaptive L1-norm principal-component analysis with online outlier rejection[END_REF], L1-IRW [START_REF] Liu | L1-subspace tracking for streaming data[END_REF],

OLP-RPCA [START_REF] Quach | Non-convex online robust PCA: Enhance sparsity via p -Norm minimization[END_REF], and OSTP [START_REF] Jia | Online Schatten quasi-norm minimization for robust principal component analysis[END_REF]. Most of them use a ℓ p -regularization (0 ≤ p ≤ 1) to discard the effect of outliers. However, they are not designed for missing data.

Robust Subspace Tracking in the Presence of Impulsive Noise

By "impulsive", we mean it can be burst noise [START_REF] Blackard | Measurements and models of radio frequency impulsive noise for indoor wireless communications[END_REF][START_REF] Ebel | The performance of Reed-Solomon codes on a bursty-noise channel[END_REF], spherically invariant random variable (SIRV) noise [START_REF] Yao | A representation theorem and its applications to spherically-invariant random processes[END_REF][START_REF] Ollila | Complex elliptically symmetric distributions: Survey, new results and applications[END_REF], or alpha-stable noise [START_REF] Nikias | Signal Processing with Alpha-Stable Distributions and Applications[END_REF][START_REF] Georgiou | Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise[END_REF]. We note that even though these algorithms were described to reduce the effect of impulsive noise in general, most simulation results were shown for burst noise only. RST algorithms that are robust to impulsive noise are summarized in Table 2.3.

Robust Variants of PAST

To take into account impulsive noise, some methods proposed in the literature have mainly been based on robust statistics so far. Among them, some studies have proposed robust variants of PAST to deal with impulsive noise. In [START_REF] Chan | A robust past algorithm for subspace tracking in impulsive noise[END_REF], a robust PAST (RPAST) was proposed. The algorithm first detects the occurrence of the impulsive noise based on a threshold, and then eliminates undesirable effects by discarding contaminated observations. The threshold is determined based on an empirical function of noise variance with the assumption that error vectors follow a Gaussian distribution corrupted by additive impulsive noise. in the presence of non-Gaussian noise with a heavy-tailed distribution. In particular, bounded nonlinear maps were employed to discard the effect of impulsive noise. Accordingly, a new robust PAST algorithm based on BNC was derived.

Adaptive Kalman Filtering

Another good approach capable of handling impulsive noise is based on adaptive Kalaman filtering. In [START_REF] Liao | A new robust Kalman filter-based subspace tracking algorithm in an impulsive noise environment[END_REF], Liao et al. proposed a RST algorithm based on an adaptive Kalman filter with variable number of measurements (KFVM). The main idea of using the KFVM is to deal with the tracking of fast-varying subspace [START_REF] Chan | A new adaptive Kalman filter-based subspace tracking algorithm and its application to DOA estimation[END_REF]. More concretely, when the underlying subspace varies quickly, a small number of past observations are exploited in the recursion and vice versa. To handle the impulsive noise, the M-estimate technique is incorporated into the KFVNM algorithm.

The complexity of the proposed KFVM-based algorithm is much higher than the PAST-based algorithms especially when the number of observations used for subspace update is large.

ROBUST SUBSPACE TRACKING IN THE PRESENCE OF COLORED NOISE

Weighted Recursive Least-Squares Method

Recently, based on robust statistics but different from the common two-step scheme mentioned above, we proposed in [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] an RST algorithm with linear computational complexity based on a weighted RLS approach, namely ROBUSTA. On the theoretical aspect, we provided a converge analysis of ROBUSTA in the presence of SIRV noise. Interestingly, we showed that it also corresponded to adaptive robust covariance estimation. ROBUSTA outperformed many stateof-the-art algorithms for burst noise, SIRV noise, and alpha-stable noise. Also, it can be easily adapted, in conjunction with pre-processing steps, to handle alpha-stable noise.

Robust Subspace Tracking in the Presence of Colored Noise

In the literature, RST algorithms that are robust to colored noise can be categorized into two groups: (i) instrumental variable and (ii) oblique projection. We summarize these algorithms in Table 2.5.

Instrumental Variable based Algorithms

For colored noise, one of the main directions is to use the instrumental variable (IV) which allows avoiding biased estimate. An appealing benefit of this approach is easy to adapt derivation from classical ST algorithms. While having improved performance, the computational complexity of IV-based algorithms is often higher than the original ones due to the selection of the IV vector size. Specifically, in [START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF], two direct extensions of the PAST algorithms, named IV-PAST and extended IV-PAST, were proposed. It is shown that their performance is enhanced, comparing to the original ones. With the aim to improve further performance in subspace-based system identification applications, several algorithms in conjunction with using IV were addressed in [START_REF] Mercère | Propagator-based methods for recursive subspace model identification[END_REF].

The key idea is to adapt the propagator approach by exploiting the relationship between array signal processing and subspace identification.

Very recently, Chan et al. in [START_REF] Chan | A new variable forgetting factor and variable regularized square root extended instrumental variable PAST algorithm with applications[END_REF] proposed a new robust variant of PAST capable of handing linear models with complex coefficients, multiple outputs, and colored noises. In the proposed method, the authors used a new adaptive forgetting factor and imposed a ℓ 2 -norm regularization into the objective function of PAST. In particular, the adaptive forgetting factor was obtained at each time instant by minimizing the mean-square deviation of the estimator from an extended IV linear model and IV-PAST. The additional ℓ 2 -norm regularized term on the weight vectors is aimed to reduce the error variance and prevent the ill-conditioned computation at low SNR levels. Generally, if low computational complexity is concerned, IV-based methods require a IV vector uncorrelated with the noise which is not always met in practice.

Oblique Projection based Algorithms

Another direction, which can avoid the above drawback, is based on oblique projection onto the subspace manifold, such as [START_REF] Chen | Subspace tracking in colored noise based on oblique projection[END_REF][START_REF] Yger | Oblique principal subspace tracking on manifold[END_REF]. It is due to the fact that the noise vector may lie in a low dimension subspace instead of being treated as full rank in the observation space. Naturally, oblique projections arise in the solution to recover the signal. Accordingly,

Sparse Subspace Tracking

Recently, sparse subspace estimation and tracking have been attracted more attention from the signal processing community due to the fact that many modern datasets admit sparse representation has huge potential capabilities for analyzing them [START_REF] Zhang | A survey of sparse representation: Algorithms and applications[END_REF]. Although several algorithms have been introduced for sparse subspace estimation in the batch setting (see [START_REF] Cai | Sparse PCA: Optimal rates and adaptive estimation[END_REF][START_REF] Papailiopoulos | Sparse PCA through low-rank approximations[END_REF][START_REF] Vu | Fantope projection and selection: A near-Optimal convex relaxation of sparse PCA[END_REF] for examples), there exist only a few studies on sparse ST algorithms so far.

In [START_REF] Wang | Online learning for sparse PCA in high dimensions: Exact dynamics and phase transitions[END_REF], Chuang and Yue proposed an adaptive algorithm called OIST (which stands for Oja's algorithm with Iterative Soft Thresholding) for online sparse PCA. The authors investigated a rank-one spiked model in a high-dimension regime and indicated that the estimate of the eigenvector from the sample covariance matrix is inconsistent. To alleviate it, they introduced an extended version of Oja's algorithm followed by a soft-thresholding step to promote sparsity on the estimate. The asymptotic convergence, steady state, and phase transition of OIST were also derived to understand its behavior in a high-dimension regime when the dimension is much larger than the number of observations. However, OIST is designed for only rank-one subspaces, i.e. lines. In parallel, a novel online sparse PCA algorithm able to deal with rank-k spiked models (k ≥ 1) was proposed via row truncation technique in [START_REF] Yang | Streaming Sparse Principal Component Analysis[END_REF]. More concretely, a simple ℓ 2 -norm based row truncation operator was introduced to zero out rows whose leverage score is below a predefined threshold. At each time instant, the QR decomposition of the resulting truncated covariance matrix was realized to update the principal subspace. The authors also proved that the proposed algorithm is consistent in the high-dimension regime.

In [START_REF] Yang | Fast STAP method based on PAST with sparse constraint for airborne phased array radar[END_REF], Xiaopeng et al. introduced a new robust variant of PAST called ℓ 1 -PAST. Specifically, the authors modified the cost function of PAST by adding a ℓ 1 -norm constraint imposed on the subspace matrix to control its sparsity. Accordingly, a new RLS algorithm like PAST was derived to minimize the proposed objective function in an efficient way. The ℓ 1 -PAST is robust and stable even when the number of observations is small.

In [START_REF] Giampouras | Online sparse and low-rank subspace learning from incomplete data: A Bayesian view[END_REF], Giampouras et al. developed a novel robust sparse ST method namely OVBSL in the lens of Bayesian inference. To deal with the sparsity constraint on the subspace matrix, OVBSL utilized the group-sparsity inducing the convex ℓ 2 /ℓ 1 -norm. Since it belongs to the family of Bayesian methods, no fine-tuning parameter is required and the proposed algorithm is fully automated.

CONCLUSIONS

In this topic, we also proposed several two-stage approach based algorithms for sparse ST in [START_REF] Lassami | Low cost subspace tracking algorithms for sparse systems[END_REF][START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF][START_REF] Lassami | Fast sparse subspace tracking algorithm based on shear and givens rotations[END_REF]. The main steps of the two-stage approach is as follows. We first utilize a well-known ST algorithm from the literature (e.g. PAST or FAPI) to extract an orthonormal basis of the underlying subspace. Then, we estimate a sparse weight matrix based on some criteria on sparsity such that it can span the same subspace. For example, in [START_REF] Lassami | Low cost subspace tracking algorithms for sparse systems[END_REF], two new algorithms SS-OPAST and DS-OPAST were designed for sparse system matrix and sparse source signals respectively.

We particularly exploited the natural gradient to find the sparsest matrix from the estimated orthonormal matrix by OPAST. In [START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF][START_REF] Lassami | Fast sparse subspace tracking algorithm based on shear and givens rotations[END_REF], we used FAPI in the first stage and then derived SS-FAPI, orthogonal SS-FAPI, and GSS-FAPI algorithms. Specifically, the sparsity criterion considered there is differentiable and smoother than the previous one in [START_REF] Lassami | Low cost subspace tracking algorithms for sparse systems[END_REF]. Accordingly, it facilitates the optimization by employing the Newton method and Taylor expansions. To sum up, a performance comparison among these sparse ST algorithms is given in Table 2.5.

Conclusions

ST has shown an increased interest in signal processing with the aim of analysing real-time big data problems and its improvement is in parallel to recent advances in optimization. In this chapter, we provided a brief survey on adaptive algorithms for RST which were mostly developed over the last decade. We highlighted three classes of RST algorithms for dealing with non-Gaussian noises including sparse outliers, impulsive noise, and colored noise. The last decade has also witnessed the widespread of high-dimensional data analysis in which sparse representation-based methods are successfully applied to many signal processing applications.

Accordingly, sparse ST algorithms are also reviewed in this chapter.

Chapter 3

Robust Subspace Tracking with

Missing Data and Outliers

INTRODUCTION

In this chapter, we propose a novel algorithm, namely PETRELS-ADMM, to deal with subspace tracking in the presence of outliers and missing data. The proposed approach consists of two main stages: outlier rejection and subspace estimation. In the first stage, alternating direction method of multipliers (ADMM) is effectively exploited to detect outliers affecting the observed data. In the second stage, we propose an improved version of the parallel estimation and tracking by recursive least squares (PETRELS) algorithm to update the underlying subspace in the missing data context. We then present a theoretical convergence analysis of PETRELS-ADMM which shows that it generates a sequence of subspace solutions converging to the optimum of its batch counterpart. The effectiveness of the proposed algorithm, as compared to state-of-the-art algorithms, is illustrated on both simulated and real data.

Introduction

Subspace estimation plays an important role in signal processing with numerous applications in wireless communications, radar, navigation, image/video processing, biomedical imaging, etc. [START_REF] Tulay | Adaptive Signal Processing: Next Generation Solutions[END_REF], especially processing modern datasets in today's big and messy data [START_REF] Vaswani | Rethinking PCA for modern data sets: Theory, algorithms, and applications[END_REF]. It corresponds to estimating an appropriate r-dimensional subspace U of R n where r ≪ n, from a set of m observed data vectors {x i } m i=1 , or equivalently, a measurement data matrix X of size n × m. To this end, the standard approach is to solve an eigen-problem in a batch manner where the underlying subspace can be obtained from either singular value decomposition of the data matrix or eigenvalue decomposition of its covariance matrix. In certain online or large-scale applications, batch algorithms become inefficient due to their high computational complexity, O(nm min(m, n)), and memory cost, O(nm) [START_REF] Golub | Matrix Computations[END_REF].

In the signal processing literature, several good surveys of the standard algorithms for subspace tracking can be found, e.g., [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF][START_REF] Tulay | Adaptive Signal Processing: Next Generation Solutions[END_REF]. The algorithms can be categorized into three classes in terms of their computational complexity: high complexity O(n 2 r), moderate complexity O(nr 2 ) and low complexity O(nr). Note that, there usually exists a trade-off among estimation accuracy, convergence rate and computational complexity. However, the performance of standard algorithms may be degraded significantly if the measurement data are corrupted by even a small number of outliers or missing observations [START_REF] Jolliffe | Principal component analysis: A review and recent developments[END_REF]. Recent surveys [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF][START_REF] Balzano | Streaming PCA and Subspace Tracking: The Missing Data Case[END_REF][START_REF] Wang | Subspace estimation from incomplete observations: A high-dimensional analysis[END_REF] show that missing data and outliers are ubiquitous and more and more common in the big data regime. This has led to attempts to define robust variants of subspace learning, namely robust subspace tracking (RST), or online robust PCA. In this work, we aim to investigate the RST problem in the presence of both outliers and missing data.

Our study is also motivated by several emerging applications in diverse fields. In big data analysis, subspace tracking is used to monitor dynamic cardiac magnetic resonance imaging (MRI), track network-traffic anomalies [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF] or mitigate radio frequency interference (RFI) in radio astronomy [START_REF] Dung | Generalized minimum noise subspace for array processing[END_REF]. Moreover, in 5G wireless communication, subspace tracking have recently 3.1. INTRODUCTION been exploited for channel estimation in massive MIMO [START_REF] Haghighatshoar | Low-complexity massive MIMO subspace estimation and tracking from low-dimensional projections[END_REF] and millimeter wave multiuser MIMO [START_REF] Buzzi | Subspace tracking and least squares approaches to channel estimation in millimeter wave multiuser MIMO[END_REF].

Related Works

In the literature, there have been several studies on subspace tracking in the missing data context. Among them, Grassmannian rank-one update subspace estimation (GROUSE) [START_REF] Balzano | Online identification and tracking of subspaces from highly incomplete information[END_REF] is an incremental gradient subspace algorithm that performs the stochastic gradient descent on the Grassmannian manifold of the r-dimensional subspace. It belongs to the class of low complexity and its convergence has recently been proved in [START_REF] Zhang | Global convergence of a Grassmannian gradient descent algorithm for subspace estimation[END_REF]. A robust version of GROUSE for handling outliers is Grassmannian robust adaptive subspace tracking (GRASTA) [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF]. GRASTA first uses an ℓ 1 -norm cost function to reduce the effect of sparse outliers and then performs the incremental gradient on the Grassmannian manifold of subspace U in a similar way as in GROUSE. Although GRASTA is one of the fastest RST algorithms for handling missing data corrupted by outliers, convergence analysis of this algorithm is not available.

Parallel estimation and tracking by recursive least squares (PETRELS) [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF] can be considered as an extension of the well-known projection approximation subspace tracking (PAST) algorithm [START_REF] Yang | Projection approximation subspace tracking[END_REF] in order to handle missing data. Specifically, PETRELS is a recursive least squares-type algorithm applying the second order stochastic gradient descent to the cost function. Inspired by PETRELS, several variants have been proposed to deal with missing data in the same line such as [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF][START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF][START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF]. The subspace tracking algorithm in [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF] is derived from minimizing the sum of squared residuals, but adding a regularization of the nuclear norm of subspace U. Robust online subspace estimation and tracking (ROSETA) in [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF] applies an adaptive step size at the stage of subspace estimation to enhance the convergence rate. Meanwhile the main idea of PETRELS-CFAR algorithm [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] is to handle "outliers-removed" data (i.e., outliers are first removed before performing subspace tracking) using a constant false alarm rate (CFAR) detector. However, the convergence of these PETRELS-based algorithms has not been mathematically proved yet.

Recursive projected compressive sensing (ReProCS)-based algorithms [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF][START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF] are also able to adaptively reconstruct a subspace from missing observations. They provide not only a memoryefficient solution, but also a precise subspace estimation as compared to the state-of-the-arts. However, they require strong assumptions on subspace changes, outlier magnitudes and accurate initialization.

Other subspace tracking algorithms, able to deal with missing data, include pROST [START_REF] Hage | Robust PCA and subspace tracking from incomplete observations using ℓ 0 -Surrogates[END_REF], APSM [START_REF]Robust subspace tracking with missing entries: The set-theoretic approach[END_REF], POPCA [START_REF] Gonen | Subspace learning with partial information[END_REF] and OVBSL [START_REF] Giampouras | Online sparse and low-rank subspace learning from incomplete data: A Bayesian view[END_REF]. They either require memorizing previous observations and good initialization or do not provide a convergence guarantee.

Among the subspace tracking algorithms reviewed above, only a few of them are robust in 3.1. INTRODUCTION the presence of both outliers and missing observations, including GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], pROST [START_REF] Hage | Robust PCA and subspace tracking from incomplete observations using ℓ 0 -Surrogates[END_REF],

ROSETA [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF], ReProCS-based algorithms [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF][START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF] and PETRELS-CFAR [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF].

Contributions

Adopting the approach of PETRELS-CFAR [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] but aiming to improve RST performance, we are interested in looking for a method that can remove outliers more effectively. Following our preliminary study presented in [START_REF] Thanh | Robust subspace tracking with missing data and outliers via ADMM[END_REF], the main contributions of the chapter are as follows.

First, we propose a novel algorithm, namely PETRELS-ADMM, for the RST problem to deal with both missing data and outliers. It includes two main stages: outlier rejection and subspace estimation and tracking. Outliers residing in the measurement data are detected and removed by our ADMM solver in an effective way. Particularly, we design an efficient augmented Lagrangian alternating direction method for the ℓ 1 -regularized loss minimization. Furthermore, we propose an improved version of PETRELS, namely iPETRELS. It is observed that PETRELS is ineffective when the fraction of missing data is too large. We thus add a regularization of the ℓ 2,∞ -norm, which aims to control the maximum ℓ 2 -norm of rows in U, in the objective function to avoid such performance loss. In addition, we introduce an adaptive step size to speed up the convergence rate as well as enhance the subspace estimation accuracy.

Second, we provide a convergence analysis of the proposed algorithm where we show that the solutions {U t } ∞ t=1 generated by PETRELS-ADMM converge to a stationary point of the expected loss function f (U) asymptotically. To the best of our knowledge, this is a pioneer analysis for RST algorithm's convergence in the presence of both outliers and missing data, under mild conditions.

Finally, we provide extensive experiments on both simulated and real data to illustrate the effectiveness of PETRELS-ADMM in three application contexts: robust subspace tracking, robust matrix completion and video background-foreground separation.

There are several differences between PETRELS-ADMM and the state-of-the-art RST algorithms. In particular, our mechanism for outlier rejection can facilitate the subspace estimation ability of RST algorithms where "clean" data involve the process only, thus improving overall performance. Excepting PETRELS-CFAR, the common principle of the state-of-the-art algorithms is "outlier-resistant" (i.e., to have a "right" direction toward the true subspace). The algorithms thus require robust cost functions as well as additional adaptive parameter selection. For examples, GRASTA and ROSETA use the ℓ 1 -norm robust estimator to reduce the effect of outliers while pROST applies the ℓ 0 -norm one instead. However, there is no guarantee that the ℓ pnorm robust estimator (i.e., p ∈ [0, 1]) can provide an optimal solution because of non-convexity.

Accordingly, the effect of outliers can not be completely removed in tracking. This is why the algorithms can fail in the appearance of a large fractions of outliers or significant subspace changes 3.2. PROBLEM FORMULATION in practice. By contrast, "detect and skip" approach like PETRELS-CFAR can utilize advantage (i.e., competitive performance) of the original PETRELS in missing observations and then treat outliers as missing data to facilitate the subspace tracking.

Compared to PETRELS-CFAR, our ADMM solver may be efficient than CFAR in terms of memory cost and flexibility. The constant false alarm rate method (CFAR) [START_REF] Shor | Performances of order statistics CFAR[END_REF] uses a moving window to detect outliers (i.e., using both old and new observations at each time instant). By contrast, our ADMM solver exploits only a new incoming data vector, hence requiring a lower storage complexity. Moreover, the performance of CFAR depends on predefined parameters such as the probability of false alarm and the size of the reference window [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF]. Our ADMM solver does not involve such parameters and hence it is more efficient. Third, PETRELS-CFAR may provide an unstable solution in the presence of a high corruption fraction due to lack of regularization (i.e., in the similar way as PETRELS).

Moreover, PETRELS-ADMM can be classified to a class of provable ST algorithms [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF][START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF] where a performance guarantee is provided. Our proposed algorithm takes both advantages of streaming solution (need only single-pass of data) and preserved convergence.

The structure of the chapter is organized as follows. Section 3.2 formulate the RST problem. Section 3.3 establishes our PETRELS-ADMM algorithm for RST and Section 3.4 gives its theoretical convergence analysis. Section 3.5 presents extensive experiments to illustrate the effectiveness of PETRELS-ADMM as compared to the state-of-the-art algorithms. Section 3.6 concludes the chapter.

Problem Formulation

Robust Subspace Tracking

Assume that at each time t, we observe a signal x t ∈ R n satisfying the following model:

x t = P t (ℓ t + n t + s t ), (3.1)
where ℓ t ∈ R n is the true signal that lies in a low dimensional subspace1 of U ∈ R n×r (i.e., ℓ t = Uw t , where w t is a weight vector and r ≪ n), n t ∈ R n is the noise vector, s t ∈ R n is the sparse outlier vector, while the diagonal matrix P t ∈ R n×n is the observation mask indicating whether the k-th entry of x t is observed (i.e., P t (k, k) = 1) or not (i.e., P t (k, k) = 0). For the sake of convenience, let Ω t be the set of observed entries at time t.

Before introducing the RST formulation, we first define a loss function ℓ(.) that remains 3.2. PROBLEM FORMULATION convex while still promoting sparsity: For a fixed subspace U ∈ R n×r and a signal x ∈ R n under an observation mask P, the loss function ℓ(U, P, x) with respect to U and {P, x} is derived from minimizing the projection residual on the observed entries and accounting for outliers as ℓ(U, P, x)

∆ = min s,w l(U, P, x, w, s) (3.2)
with l(U, P, x, w, s) = P(Uw + sx)

2 2 + ρ ∥s∥ 1 , (3.3)
where we here use the ℓ 1 regularization to promote entry-wise sparsity on s and ρ > 0 is a regularization parameter to control the degree of the sparsity. 2Now, given a streaming set of observed signals, X = {x i } t i=1 in (3.1), we wish to estimate a rank-r matrix U t ∈ R n×r such that it can cover the span of the complete-data noiseless signal ℓ t .

RST can be achieved via the following minimization problem:

U t = argmin U∈R n×r f t (U) ∆ = 1 t t i=1 β t-i i ℓ(U, P i , x i ) , (3.4)
where the forgetting factor β i ∈ (0, 1] is to discount the effect of past observations. For the convergence analysis, we will consider the expected cost f (U) on signals distributed by the true data-generating distribution P data , instead of the empirical cost f t (U). Thanks to the law of large numbers, expectation of the observations without discounting (i.e., β = 1) converges to the true value when t tends to infinity,

Û = argmin U∈R n×r f (U) ∆ = E x i.i.d ∼ P data [ℓ(U, P, x)] = lim t→∞ f t (U) . (3.5)
From the past estimations {s i , w i } t i=1 , instead of minimizing the empirical cost function f t (U) in (3.4), we propose to optimize the surrogate g t (U) of f t (U), which is defined as

g t (U) = 1 t t i=1 β t-i i P i (Uw i + s i -x i ) 2 2 + ρ ∥s i ∥ 1 , (3.6)
where {s i , w i } t i=1 are considered as constants. Note that, the surrogate function provides an upper bound on f t (U). In our convergence analysis, we will prove that f t (U t ) and g t (U t )

converge almost surely to the same limit. As a result, the solution U t obtained by minimizing g t (U) is exactly the solution of f t (U) when t tends to infinity.

Assumptions

We make the following assumptions for convenience of convergence analysis as well as helping deploy our optimization algorithm:

(A-1): The data-generation distribution P data has a compact support, x i.i.d ∼ P data . Indeed, real data are often bounded such as audio, image and video, hence this assumption naturally holds in many situations.

(A-2): U is constrained to the set

U ∆ = {U ∈ R n×r , ∥U :,k ∥ 2 ≤ 1, 1 ≤ κ(U) ≤ α} with a constant α.
The first constraint ∥U :,k ∥ 2 ≤ 1 is not restrictive as it is considered to bound the scale of basis vectors in U and hence prevents the arbitrarily very large values of U. While the low condition number of the subspace κ(U) is to prevent the ill-conditioned computation.

(A-3): Coefficients w are constrained to the set

W = {w ∈ R r , ω 1 ≤ |w(i)| ≤ ω 2 , i = 1, 2, . . . , r}
with two constants ω 1 and ω 2 , 0 ≤ ω 1 < ω 2 . Since the data x and subspace U are assumed to be bounded, it is natural that the subspace weight vector w is bounded too.

(A-4): The subspace changes at two successive time instances is small, i.e., the largest principal angle between U t and U t-1 is 0 ≤ θ max ≪ π/2, or the distance between the two subspaces,

d(U t , U t-1 ) = sin(θ max ), satisfies 0 ≤ d(U t , U t-1 ) ≪ 1.

Proposed PETRELS-ADMM Algorithm

In this section, we present a novel algorithm, namely PETRELS-ADMM, for RST to handle missing data in the presence of outliers. The main idea is to minimize the empirical cost function g t in (3.6) by updating outliers s t , weight vector w t and subspace U t alternatively.

Under the assumption (A-2) that the underlying subspace U changes slowly, we can detect outliers in s t by projecting the new observation x t into the space spanned by the formerly estimated subspace U t-1 in the previous phase. Specifically, we solve the following minimization problem:

{s t , w t } ∆ = argmin s,w l(U t-1 , P t , x t , w, s) (3.7) with l(U t-1 , P t , x t , w, s) = P t (U t-1 w + s -x t ) 2 2 + ρ ∥s∥ 1 . (3.8)
In the second phase, the subspace U t can be estimated by minimizing the sum of squared residuals:

U t = argmin U 1 t t i=1 β t-i tr( P i ) n P i (Uw i -x i ) 2 2 + α 2t ∥U∥ 2 2,∞ , (3.9
aaaa{s i , w i } = argmin s,w P i (U i-1 w + s -x i ) 2 2 + ρ ∥s∥ 1 .
// Update the new mask P i :

aaaa P i (k, k) = P i (k, k), if s i (k) = 0, P i (k, k) = 0, otherwise.
// Estimate subspace U i using Algorithm 3:

aaaaU i = argmin U 1 i i j=1 β i-j tr( P j ) n P j (x j -Uw) 2 2 + α 2i ∥U∥ 2 2,∞ .
end for

Output: U t ∈ R n×r
where the regularization α 2t ∥U∥ 2 2,∞ is to bound the scale of vectors in U while the outliers s t has been disregarded and the new observation P i are determined by the following rule:

   P i (k, k) = P i (k, k), if s i (k) = 0, P i (k, k) = 0, otherwise, (3.10)
which we aim to skip the corrupted entries of x i .

Our algorithm first applies the ADMM framework in [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF], which has been widely used in previous works for solving (3.7), and then propose a modification of PETRELS [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF] to handle (3.9).

In the outlier rejection stage, we emphasize here that we propose to focus on augmenting s (as shown in (3.12)) to further annihilate outlier effect, unlike GRASTA and ROSETA which focus on augmenting the residual error only. 3 Meanwhile, we modify the subspace update step in PETRELS by adding an adaptive step size η t ∈ (0, 1] at each time instant t, instead of a constant one as in the original version. The modification can be interpreted as an approximation of Newton method. The proposed method is summarized in Algorithm 1. 3 In GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF] and ROSETA [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF], both the authors aimed to detect outliers s by solving the augmented Lagrangian of (3.7) as follows

L(s, y, w) = ∥s∥ 1 + y ⊤ Pt(Ut-1w + s -xt) + ρ 2 Pt Ut-1w + s -xt 2 2 .

PROPOSED PETRELS-ADMM ALGORITHM

Algorithm 2: Outlier Detection Input: Observed signal x t ∈ R n×1 , observation mask P t ∈ R n×n , the previous estimate U t-1 ∈ R n×r , maximum iteration K, penalty parameters ρ 1 , ρ 2 , absolute and relative tolerances ϵ abs and ϵ rel . Initialization:

• Choose {u 0 , s 0 , w 0 , z 0 , e 0 } randomly.

• {r 0 , e 0 } ← 0 n

Main Program:

Procedure:

for k = 0, 1, . . . , K Cost // Update w w k+1 = (P t U i-1 ) # P t (x t -s k + e k ) 2Ω t r 2 + Ω t r z k+1 = P t (U t-1 w k+1 + s k -x t ) Ω t r e k+1 = ρ2 1+ρ2 z k+1 + 1 1+ρ2 S 1+ 1 ρ 2 (z k+1 ) Ω t // Update s u k+1 = 1 1+ρ1 P t (x t -U t-1 w k+1 ) -ρ 1 (s k -r k ) Ω t r s k+1 = S ρ/ρ1 (u k+1 + r k ) Ω t r k+1 = r k + u k+1 -s k+1 Ω t // Stopping criteria if s k+1 -s k 2 < √ nϵ abs + ϵ rel ρ 1 r k+1 2 break; Ω t end if
end for Output: s, w

Online ADMM for Outlier Detection

We show in the following how to solve (3.7) step-by-step:

Update s t
To estimate outlier s t given w, we exploit the fact that (3.7) can be cast into the ADMM form as follows:

min u,s h(u) + q(s) subject to u -s = 0, (3.11)
where u is the additional decision variable, h(u) = 1 2 ||P t (U t-1 w + ux t )|| 2 2 and q(s) = ρ∥s∥ 1 . The corresponding augmented Lagrangian with the dual variable vector β is thus given by

L(s, u, β) = q(s) + h(u) + β ⊤ (u -s) + ρ 1 2 ∥u -s∥ 2 2 , (3.12)
where ρ 1 > 0 is the regularization parameter 4 . Let r = β/ρ 1 be the scaled dual variable, we can rewrite the Lagrangian (3.12) as follows:

L(s, u, r) = q(s) + h(u) + ρ 1 r ⊤ (u -s) + ρ 1 2 ∥u -s∥ 2 2 . (3.13)
The optimization of (3.13) is achieved iteratively where we have the following update rule using the scaled dual variable at the k-th iteration,

u k+1 = argmin u h(u) + ρ 1 (r k ) ⊤ (u -s k ) + ρ 1 2 u -s k 2 2 , (3.14)
s k+1 = argmin s q(s) -ρ 1 (r k ) ⊤ s + ρ 1 2 u k+1 -s 2 2 , (3.15)
r k+1 = r k + u k+1 -s k+1 . (3.16)
In particular, we first exploit that the minimization (3.14) can be formulated as a convex quadratic form:

u k+1 = argmin u 1 + ρ 1 2 ∥u∥ 2 2 -P t (x t -U t-1 w) -ρ 1 (s k -r k ) ⊤ u = 1 1 + ρ 1 P t (x t -U t-1 w) -ρ 1 (s k -r k ) . (3.17)
Meanwhile, (3.15) is a standard proximal minimization with the ℓ 1 -norm [START_REF] Parikh | Proximal Algorithms[END_REF] as

s k+1 = argmin s ρ ∥s∥ 1 + ρ 1 2 s -u k+1 + r k 2 2 = S ρ/ρ 1 u k+1 + r k , (3.18)
where S a (x) is a thresholding operator applied element-wise and defined as

S a (x) =          0, if |x| ≤ a, x -a, if x > a, x + a, if x < -a, (3.19)
which is a proximity operator of the ℓ 1 -norm. Finally, a simple update rule for the scaled dual variable r can be given by the dual ascent, as

r k+1 = r k + γ k ∇ L (r k ), (3.20)

PROPOSED PETRELS-ADMM ALGORITHM

where the gradient ∇ L (r k ) is computed by ∇ L (r k ) = ρ 1 (u k+1 -s k+1 ) and γ k > 0 is the step size controlling the convergence rate. For ADMM methods, the regularization parameter is often used as the the step size for updating dual variables [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. Due to the scaled version r of the dual variable β, the step size γ k is here set to be γ k = 1/ρ 1 at the k-th iteration.

Update w t

To estimate w t given s, (3.7) can be recast into the following ADMM form:

min w∈W,e∈R n×1 1 2 P t (U t-1 w + s -x t ) 2 2 + y(e), subject to P t (U t-1 w + s -x t ) = e, (3.21)
where y(e) is a convex regularizer function for the noise e, (e.g. y(e) = σ 2 ∥e∥ 2 2 , with σ -1 can be chosen as the signal to noise ratio, SNR). The minimization (3.21) is equal to the following optimization:

min w∈W,e∈R n×1 ∥e∥ 2 2 subject to P t U t-1 w + s -x t = e. (3.22)
However, the noise e is still affected by outliers because s may not be completely rejected in each iteration. Therefore, (3.22) can be cast further into the ADMM form such that it can lie between least squares (LS) and least absolute deviations to reduce the impact of outliers. The Huber fitting can bring transition between the quadratic and absolute terms of L w,e (w, e) 5 , as

L w,e (w, e) = f Hub (e) + ρ 2 2 P t U t-1 w + s -x t -e 2 2 , (3.23)
where ρ 2 > 0 is the penalty parameter whose characteristics are similar to that of ρ 1 and the Huber function is given by [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF] f

Hub (x) =    x 2 /2, |x| ≤ 1, |x| -1/2, |x| > 1. (3.24)
As a result, e-updates for estimating w involves the proximity operator of the Huber function, that is, Cost

e k+1 = ρ 2 1 + ρ 2 P t U t-1 w k+1 + s -x t + 1 1 + ρ 2 S 1+ 1 ρ 2 P t U t-1 w k+1 + s -x t . ( 3
x t = ∥ P t x t -P t U t-1 w t ∥ 2 ∥w t ∥ 2 Ω t r η t = x t x 2 t + 1 O(1) if η t > η then η t = 1 end if O(1)
for m = 1 to n do

R m t = βR m t-1 + P t (m, m)w t w ⊤ t r 2 H m t = R m t + α 2 I r a t = (H m t ) -1 w t O(r 2 ) u m t = u m t-1 + η t ξ t P t (m, m)(x re t (m) -w ⊤ t u m t-1 )a t r end for Output: U t ∈ R n×r
Hence, at the (k + 1)-th iteration, w k+1 can be updated using the following closed-form solution of the convex quadratic function:

w k+1 = P t U t-1 # P t x t -s + e k . (3.26)
To sum up, the rule for updating w t can be given by

w k+1 = P t U t-1 # P t x t -s + e k , (3.27)
z k+1 = P t U t-1 w k+1 + s -x t , (3.28)
e k+1 = ρ 2 1 + ρ 2 z k+1 + 1 1 + ρ 2 S 1+ 1 ρ 2 z k+1 . (3.29)
We note that, by using the Huber fitting operator, our algorithm is better in reducing the impact of outliers than GRASTA and ROSETA which use ℓ 2 -norm regularization.

The procedure is stopped when the number of iterations reaches the maximum or the accuracy tolerance for the primal residual and dual norm has been met, i.e.,

s k+1 -s k 2 < √ nϵ abs + ϵ rel ρ 1 r k+1 2 , (3.30)
where ϵ abs > 0 and ϵ rel > 0 are predefined tolerances for absolute and relative part respectively.

A reasonable range for the absolute tolerance may be [10 -6 , 10 -3 ], while [10 -4 , 10 -2 ] is good for the relative tolerance, see [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF] for further details of the stopping criterion. The main steps of the outlier detection are summarized as Algorithm 2.

3.3. PROPOSED PETRELS-ADMM ALGORITHM

Improved PETRELS for Subspace Estimation

Having estimated s t , we optimize the following minimization

U t := argmin U gt (U) = 1 t t i=1 β t-i tr( P i ) n P i (x i -Uw i ) 2 2 + α 2t ∥U∥ 2 2,∞ , (3.31)
where the observation mask P i is computed by (3.10).

Thanks to the parallel scheme of PETRELS [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF], the optimal solution of the problem (3.31) can be obtained by solving its subproblems at each row u m of U, 1 ≤ m ≤ n:

u m t = argmin u m 1 t t i=1 β t-i ξ i P i (m, m) x i (m) -w ⊤ i u m 2 + α 2t ∥u m ∥ 2 2 , (3.32)
where ξ i = tr( P i ) n . In this way, we can speed up the subspace update by ignoring the u m if the m-th entry of x t is labeled as missing observation or outlier.

Thanks to Newton's method, we can update each row of U t by the following rule:

u m t = u m t-1 -H t (u m ) -1 ∂g t (U) ∂u m u m =u m t-1 , (3.33)
where the first derivative of gt is given by

∂g t (U) ∂u m = -2 t t i=1 β t-i ξ i P i (m, m) x i (m) -w ⊤ i u m w ⊤ i + α t u m , (3.34)
and the second derivative of gt , Hessian matrix, is given by

H t (u m ) = 2 t t i=1 β t-i ξ i P i (m, m)w i w ⊤ i + α t I. (3.35)
Specifically, the partial derivative ∂gt(U) ∂u m at u m t-1 can be expressed by

∂g t (U) ∂u m u m =u m t-1 = ∂g t-1 (U) ∂u m u m =u m t-1 + α t u m t-1 -u m t-2 - 2 t ξ t P t (m, m) x t (m) -w ⊤ t u m t-1 w ⊤ t . (3.36) Since u m t-1 = argmin ∂g t-1 (U) ∂u m
and the parameter α/t is small, so ∂g t-1 (U)

∂u m u m =u m t-1
= 0 and then

∂g t (U) ∂u m u m =u m t-1 ≈ -2 t ξ t P t (m, m) x t (m) -w ⊤ t u m t-1 w ⊤ t . ( 3
H m t ∆ = H ft (u m t-1 ) = 2 t R m t + α 2 I . (3.38)
Therefore, a relaxed approximation of the recursive update (3.33) can be given by

u m t ≈ u m t-1 + η t ξ t P t (m, m) x t (m) -w ⊤ t u m t-1 a ⊤ t , (3.39)
where H m t = R m t + α 2 I 6 , a t = (H m t ) -1 w t and η t denotes the adaptive step size η t ∈ [0, 1] at each time instant t, instead of a constant as in the original PETRELS [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF]. We here determine the adaptive step size η t as follows

η t = x t x 2 t + 1 with x t = ∥e t ∥ 2 ∥w t ∥ 2 , (3.40)
where the residual error e t is computed by e t = P t x t -P t U t-1 w t . Note that, the adaptive step size η t can be expressed by η t = sin(θ t ), see Fig. 3.1. The smaller angle θ t is, the closer to the true subspace we are, the smaller step size is needed. The update is summarized in Algorithm 3.

Computational Complexity Analysis

The number of floating-point operations (flop) is used to measure the computational complexity of the proposed PETRELS-ADMM. At the k-th iteration in the outlier detection phase, our method requires O(Ωr 2 ) flops where Ω is average number of observed entries at each time instant

(Ω ≤ n).
It is practically stated that the ADMM solver can converge within a few tens of iterations [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF] (also see Fig. 3.3). Therefore, the removal of outliers costs the averaged O(Ωr 2 ).

The complexity of the subspace estimation phase is also O(Ωr 2 ) as the original PETRELS [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF].

The overall computational complexity of PETRELS-ADMM is of order O(Ωr 2 ) flops.

6 H m t ∈ R r×r is a matrix of rank-one updates, so its inverse matrix can be efficiently computed recursively, thanks to Sherman-Morrison formula [START_REF] Hager | Updating the inverse of a matrix[END_REF]. Also, the small regularization parameter α > 0 can help the recursive update having a better numerical stability. The computational complexity is of order O(r 2 ).

PERFORMANCE ANALYSIS

Performance Analysis

In this section, we provide a convergence analysis for the proposed PETRELS-ADMM algorithm.

Inspired by the results of convergence of empirical processes for online sparse coding in [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF] and online robust PCA in [START_REF] Feng | Online robust PCA via stochastic optimization[END_REF][START_REF] Shen | Online optimization for max-norm regularization[END_REF], we derive a theoretical approach to analyze the convergence of values of the objective function {f t (U t )} ∞ t=1 as well as the solutions {U t } ∞ t=1 generated by PETRELS-ADMM.

Given assumptions defined in Section 3.2.2, our main theoretical result can be stated by the following theorem:

Theorem 2 (Convergence of PETRELS-ADMM). In the stationary context, let {U t } ∞ t=1 be the sequence of solutions generated by PETRELS-ADMM, then the sequence converges to a stationary point of the expected loss function f (U) when t → ∞.

Proof Sketch. Our proof can be divided into three main stages as follows: We first prove that the solutions {U t , s t } t≥1 generated by the PETRELS-ADMM algorithm are optimal w.r.t. the cost function in (3.6). We then prove that a nonnegative sequence {g t (U t )} ∞ t=1 converges almost surely where {U t } ∞ t=1 is the sequence of optimal solutions generated by the PETRELS-ADMM algorithm. After that, we prove that the surrogate {g t (U t )} ∞ t=1 converges almost surely to the empirical loss function {f t (U t )} ∞ t=1 as well as the true loss function, i.e., g t (U t ) a.s.

→ f t (U t )
a.s.

→

f (U t ), thanks to the central limit theorem.

Due to space limitation, we here present key results and report their proof sketch. The details of their proofs are provided in our appendix.

PERFORMANCE ANALYSIS

Lemma 1 (Convergence of Algorithm 2). At each time t, let {s k , u k , r k , w k , e k } ∞ k=1 be a sequence generated by Algorithm 2 for outlier detection, there always exists a set of positive numbers {c u , c s , c r , c w , c e } such that, at each iteration, the minimizers satisfy

L s k+1 , u k+1 , r k+1 , w k+1 , e k+1 ≤ L s k , u k , r k , w k , e k -c u u k -u k+1 2 2 -c s s k -s k+1 2 2 -c r r k -r k+1 2 2 -c w w k -w k+1 2 2 -c e e k -e k+1 2 2 , (3.41)
where the Lagrangian L(s, u, r, w, e) for updating these variables is a combination of two functions (3.13) and (3.23), as L s, u, r, w, e = q(s)

+ h(u) + ρ 1 r ⊤ (u -s) + ρ 1 2 ∥u -s∥ 2 2 + f Hub (e) + ρ 2 2 P t U t-1 w + s -x t -e 2 2 .
(3.42)

The asymptotic variation of s k (i.e., outliers) is then given by

lim k→∞ s k+1 -s k 2 2 = 0. (3.43)
Proof Sketch. We state the following proposition, which is in the same line as in previous convergence analysis of ADMM algorithms [START_REF] Li | Global convergence of splitting methods for nonconvex composite optimization[END_REF][START_REF] Wang | Global convergence of ADMM in nonconvex nonsmooth optimization[END_REF], used to prove the first part of lemma 1.

Proposition 1. Let {s k , u k , r k , w k , e k } ∞ k=1 be a sequence generated by Algorithm 2 and denote q k be one of these variables, the minimizer q k+1 of (3.13) satisfies

L q k+1 , . ≤ L q k , . -c q q k -q k+1 2 2 , (3.44)
where c q is a positive number.

As a result, the cluster

{s k , u k , r k , w k , e k }
m 1 ∥U t+1 -U t ∥ 2 F ≤ g t (U t+1 ) -g t (U t ) , (3.46)
m 2 ∥U t+1 -U t ∥ F ≥ g t (U t+1 ) -g t (U t ) . (3.47)
Proof Sketch. To prove that g t (U) is strongly convex, we state the following facts: g t (U) is continuous and differentiable; its second derivative is a positive semi-definite matrix (i.e., ∇ 2 U g t (U) ⪰ mI); and the domain of g t (U) is convex. In order to satisfy the Lipschitz condition, we show that the first derivative of g t (U) is bounded.

Lemma 2 (Convergence of Algorithm 3). Given an outlier vector s t generated by Algorithm 2 at each time instant t, Algorithm 3 can provide a local optimal solution U t for minimizing g t (U). Moreover, the asymptotic variation of estimated subspaces {U t } t≥1 is given by

∥U t -U t+1 ∥ F a.s. → O 1 t . ( 3

.48)

Proof. To establish the convergence, we exploit the fact that our modification can be seen as an approximate of the Newton method,

U t ∼ = U t-1 -η t H ft (U t-1 ) -1 ∇g t (U t-1 ), (3.49)
where H ft (U t-1 ) and ∇g t (U t-1 ) are the Hessian matrix and gradient of the function gt (U) at where {F t } t>0 is the filtration of the past estimations at time instant t.

U t-1 ,
m 1 ∥U t+1 -U t ∥ 2 F ≤ g t (U t+1 ) -g t (U t ) ≤ m 2 ∥U t+1 -U t ∥ F (3.50) ⇔ ∥U t -U t+1 ∥ F ≤ m 2 m 1 = O 1 t . ( 3
Proof Sketch. Let us define the indicator function δ t as follows

δ t ∆ =    1 if E g t+1 (U t+1 ) -g t (U t ) F t > 0, 0 otherwise. (3.53)
According to the quasi-martingale convergence theorem [START_REF] Bottou | Online learning and stochastic approximations[END_REF]Section 4.4], in order to show the convergence of the nonnegative stochastic process {g t (U t )} ∞ t=1 , we will prove

∞ t=0 E δ t E g t+1 (U t+1 ) -g t (U t ) F t < ∞. (3.54)
In particular, we first indicate the following inequality:

g t+1 (U t+1 ) -g t (U t ) ≤ ℓ(U t , P t+1 , x t+1 ) -f t (U t ) t + 1 . (3.55)
Since E ℓ(U t , P t+1 , x t ) = f (U t ), we have a nice property:

E g t+1 (U t+1 ) -g t (U t ) F t ≤ E ℓ(U t , P t+1 , x t+1 ) -f t (U t ) F t t + 1 = f (U t ) -f t (U t ) t + 1 . (3.56)
We then have

E δ t E g t+1 (U t+1 ) -g t (U t ) F t ≤ E √ t f (U t ) -f t (U t ) 1 √ t(t + 1) . (3.57)
Under the given assumptions, we exploit the fact that the set of measurable functions {ℓ(U i , P, x)} i≥1 defined in (3.2) is P-Donsker. Therefore, the centered and scaled version of the empirical function f t (U t ) satisfies the following proposition: -→ f t (U t ).

E √ t f (U t ) -f t (U t ) = O(1), (3.58
(3.60)

Proof Sketch. We begin the proof with providing the following inequality:

g t (U t ) -f t (U t ) t + 1 ≤ u t -u t+1 (S-1) + ℓ(U t , P t+1 , x t+1 ) -f t (U t ) t + 1 (S-2) , (3.61)
where u t ∆ = g t (U t ). We then prove that the two sequences (S-1)-(S-2) converge almost surely.

As a result, the sequence

(g t (U t ) -f t (U t ))
f t (U t ) ≤ f t (U) + L 2 ∥U -U t ∥ 2 F , ∀ U ∈ R n×r , (3.63)
where L is a positive constant. In other words, U t is the minimum point of f (U).

EXPERIMENTS

Proof Sketch. Let us denote the error function e t (U) = g t (U) -f t (U).

Due to g t (U t ) a.s.

→ f t (U t ) when t → ∞, we have ∇e t (U t ) = 0 and hence the following inequality

∥∇e t (U)∥ ≤ L 2 ∥U -U t ∥ F . (3.64)
It is therefore that

|e t (U) -e t (U t )| ∥U -U t ∥ F ≤ L 2 ∥U -U t ∥ F , (3.65)
thanks to the mean value theorem. In other word, we have

|e t (U)| ≤ L 2 ∥U -U t ∥ 2 F because of e t (U t ) a.s. → 0.
In addition, for all U ∈ R n×r , we always have f t (U t ) ≤ g t (U). Therefore, we can conclude the corollary as follows

f t (U t ) ≤ g t (U t ) = f t (U) + e t (U) ≤ f t (U) + L 2 U -U t 2 F . (3.66)
It ends the proof.

Experiments

In this section, we evaluate the performance of the proposed algorithm by comparing it to the state-of-the-art in three scenarios relative to: robust subspace tracking, robust matrix completion and video background-foreground separation respectively. In particular, extensive experiments on simulated data are conducted to demonstrate the convergence and robustness of our PETRELS-ADMM algorithm for subspace tracking and matrix completion. While four real video sequences are used to illustrate the effectiveness of PETRELS-ADMM for background-foreground separation.

Robust Subspace Tracking

In the following experiments, data x t at each time t is generated randomly using the standard signal model as in (3.1)

x t = P t (Uω t + n t + s t ), (3.67)
where U ∈ R n×r denotes a mixing matrix, ω t is a random vector living on R r space (i.e., ℓ t = Uω t ) and they are Gaussian i.i.d. of pdf N (0, 1); n t represents the white Gaussian noise N (0, σ 2 ), with SNR = -10 log 10 (σ 2 ) is the signal-to-noise ratio to control the impact of noise 3.5. EXPERIMENTS on algorithm performance; and s t is uniform i.i.d. over [0, fac-outlier] given the magnitude fac-outlier of outliers that aim to create a space for outliers. Indices of missing entries and outliers are generated randomly using the Bernoulli model with the probability ω missing and ω outlier respectively. The two probabilities represent the density of missing entries and outliers in the data.

In order to evaluate the subspace estimation accuracy, we use the subspace estimation performance (SEP) [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] metric

SEP = 1 L L i=1 tr U # es-i (I -U ex U # ex )U es-i tr U # es-i (U ex U # ex )U es-i , (3.68)
where L is the number of independent runs, U ex and U es-i are the true and the estimated subspaces at the i-th run respectively. Particularly, the denominator measures the sum of the squares of the cosines of the principal angles between U es-i and U ex , while the numerator evaluates the similar sum but for the two subspaces U es-i and the orthogonal complement U ⊥ ex . Accordingly, the lower SEP is, the better the algorithm performance is.

State-of-the-art algorithms for comparison are: GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], ROSETA [START_REF] Mansour | A robust online subspace estimation and tracking algorithm[END_REF] and PETRELS-CFAR [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF], ReProCS [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF] and NORST [START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF]. Throughout our experiments, their algorithm parameters are set by default as mentioned in the algorithms. In particular, we set a penalty parameter ρ = 1.8 and a constant step-size scale C = 2 in GRASTA. An adaptive step size of ROSETA is initialized at µ 0 = C 1+η 0 with C = 8 and η 0 = 99, while two thresholds for controlling the step size are set at η low = 50 and η high = 100. PETRELS-CFAR includes a forgetting factor set at λ = 0.999, a window size N w = 150 and a false alarm probability P fa varied from [0.1, 0.7] depended on the outlier intensity. Both ReProCS and NORST require several predefined parameters, including t train = 200 data samples, α = 60, K = 33 and ω eval = 7.8 × 10 -4 . For our algorithm, we set the penalty parameters at 1.5, the regularization parameter α = 0.1 and the step-size threshold η = sin(π/3), while the maximum number of iterations for outlier detection phase is fixed at K = 50. Matlab codes are available online 7 . The experimental results are averaged over 100 independent runs.

Convergence of PETRELS-ADMM

To demonstrate the convergence of our algorithm, we use a synthetic data whose number of row n = 50, rank r = 2, and 5000 vector samples with 90% entries observed on average. Specifically, the outlier density ω outlier is varied from 0.05 to 0.4, while the outlier intensity is set at three and the weight ρ. We can see that, the variation of {s k } k≥1 always converges in all testing cases.

When the penalty parameter ρ ≥ 0. → O(1/t) almost surely.

Outlier Detection

Following the above experiment, we next assess the ability of PETRELS-ADMM for outlier detection against the noise level. The three statistical metrics including Sensitivity (SEN) and Specificity (SEP) and Accuracy (ACC) are used to evaluate its outlier detection performance [START_REF] Powers | Evaluation: From precision, recall and F-measure to ROC, informedness, markedness and correlation[END_REF].

Particularly, SEN measures the percentage of outliers detected correctly over the total outliers in the measurement data. SEP is similar to SEN, but for normal entries and ACC indicates how the estimator makes the correct detection. We use the same data above, but 20% of the observations are missing. The outlier density ω outlier is set at 0.2, while two intensity levels are considered, with fac-outlier ∈ {1, 10}. As can be seen that when we increase the value of SNR from -20 dB to 20 dB, the detection accuracy goes up first and then converges towards a constant level. At very low SNRs (i.e., < 0 dB), the proposed algorithm does not work well in which many normal entries are labeled as outliers, although the number of correctly detected outliers are high. When SNR > 0 dB, PETRELS-ADMM achieves a competitive prediction accuracy with respect to all three evaluation metrics. Fig. 3.5 provides more practical evidences to demonstrate the effectiveness of PETRELS-ADMM for the outlier detection. Particularly, the locations of outliers s t are well detected even when the measurement data is corrupted by noise with a moderate SNR value (e.g. 10 dB). Also, amplitude of the outliers is recovered nearly correctly with a small relative error (RE = ∥st-st∥ 2 ∥st∥ 2 ) in both cases (e.g. RE = 0.0616 at the 20 dB noise level). As a result, the corrupted signals are also well reconstructed, as illustrated in Fig. 3.5(b) and (d).

Robustness of PETRELS-ADMM

To investigate the robustness of PETRELS-ADMM, we vary the outlier intensity, density and missing density and then measure the SEP metric. Moreover, we also demonstrate the effectiveness of PETRELS-ADMM against noisy and time-varying environments.

Impact of outlier intensity on algorithm performance

We fix n = 50, r = 2, 90% entries observed, outlier density ω outlier = 0. values. At low outlier intensity (i.e., fac-outlier ≤ 1), all algorithms yield good accuracy with fast convergences, though ROSETA and ReProCS obtain the higher SEP (i.e., ≈ 10 -3 ) as compared to that of the four remaining algorithms. In particular, PETRELS-ADMM provides the best subspace estimation accuracy, i.e., SEP ≈ 10 -5 in both cases (see Fig. PETRELS-ADMM outperforms the four remaining algorithms in this context. In particular, our algorithm performs very well even when the fraction of outliers is high (e.g. ω outlier = 0.4).

By contrast, four algorithms including GRASTA, ROSETA, ReProCS and NORST may fail to track subspace in the case of a high outlier density (see Fig.

Impact of the density of missing entries on algorithm performance

Following the above experiments, we change the number of missing entries in the measurement data by varying the probability ω missing while fixing the other attributes. The results are reported in Fig. 7.13 and Fig. 3.9. In particular, the effect of ω missing on algorithm performance is presented in Fig. 7.13. Similarly, PETRELS-ADMM yields the best performance in four cases of missing observations. Three algorithms including PETRELS-CFAR, GRASTA and ReProCS provide good performance but with slower convergence rate and accuracy, while ROSETA and NORST have failed again in this task due to the high outlier intensity (i.e., fac-outlier = 10). As can be seen from Fig. 3.9(a)-(c) that the state-of-the-art algorithms only perform well when the number of corruptions is smaller than half the number of entries in the data measurement.

While PETRELS-ADMM still obtains the reasonable subspace estimation performance in terms of SEP (i.e., ≈ 10 -3 ) in the case of very high corruptions, see Fig. 3.9(d).

Noisy and Time-Varying Environments

We first investigate the effect of the noise on the performance of PETRELS-ADMM in comparison with the state-of-the-art algorithms. We vary the value of SNR in the range from dB to 20 dB and assess their performance on the same data above. Experimental results are illustrated in Fig. 3.10. As can be seen that the convergence rate of PETRELS-ADMM is not affected by varying environments. Particularly, the true subspace U is supposed to be varying with time under the model U t = (1 -ε)U t-1 + εN t , where N t ∈ R n×r is a Gaussian noise matrix (zero-mean and unit-variance) and ε is to control the subspace change which is chosen among {10 -1 , 10 -2 , 10 -3 }. We use the same signal model as in the previous tasks and 1000 vector samples. Also, we create an abrupt change at t = 500 to see how fast the proposed algorithm can converge. We measure the performance of PETRELS-ADMM at two noise levels (SNR = 5 and 10 dB) with different corruption fractions. Experimental results are illustrated in Fig. 7.12(a)-(d). In the same manner to the effect of the noise, the time-varying factor ε does not affect the convergence rate of PETRELS-ADMM, but only its subspace estimation. Fig. 7.12 shows that the estimation accuracy of the proposed algorithm will decrease if the time-varying factor ε increases. When the underlying subspace varies slowly (e.g. ε ≤ 10 -2 ), the resulting values of SEP, which always converge towards an error floor, indicate that PETRELS-ADMM can be robust to slowly time-varying scenarios.

Robust Matrix Completion

We compare here the robust matrix completion (RMC) performance using PETRELS-ADMM with GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], LRGeomGC [START_REF] Vandereycken | Low-rank matrix completion by Riemannian optimization[END_REF] and RPCA-GD [START_REF] Yi | Fast algorithms for robust PCA via gradient descent[END_REF].

The measurement data X = P ⊛ (UW + S + N) used for this task corresponds to the rank-2 matrices of size of 400 × 400, where the operator ⊛ denotes the Hadamard product. Particularly, we generated the mixing matrix U ∈ R 400×2 and the coefficient matrix W ∈ R 2×400 at random.

Their entries were random variables that follow Gaussian distribution with zero mean and unit variance. The measurement data X was corrupted by a white Gaussian noise N ∈ R 400×400

whose SNR is fixed at 40 dB. In the literature, the SNR value of around 40 dB is used for performance evaluation of completion algorithms due to missing observations and/or outliers at low-noise conditions [START_REF] Nguyen | Low-rank matrix completion: A contemporary survey[END_REF]. The data matrix was affected by different percentages of missing (P) and outliers (S) from 0% -90%. The location and value of corrupted entries (including missing and outliers) were uniformly distributed. when the data is corrupted by a very high corruption fraction. At high outlier intensity (i.e., fac-outlier ≥ 1), PETRELS-ADMM based RMC provides the best matrix reconstruction error performance, GRASTA still retain good performance, while RPCA-GD and LRGeomGC fail to recover corrupted entries.

Video Background/Foreground Separation

We further illustrate the effectiveness of the proposed PETRELS-ADMM algorithm in the application of RST for video background/foreground separation, and compare with GRASTA and PETRELS-CFAR. We use four real video sequences for this task, including Hall, Lobby, Sidewalk and Highway datasets. In particular, the two former datasets are from GRASTA's homepage8 , while the two latter datasets are from CD.net2012 3.6. CONCLUSIONS

Conclusions

In this chapter, we have proposed an efficient algorithm, namely PETRELS-ADMM, for the robust subspace tracking problem to handle missing data in the presence of outliers. By converting the original RST problem to a surrogate one, which facilitates the tracking ability, we have derived an online implementation for outlier rejection with a low computational complexity and a fast convergence rate while still retaining a high subspace estimation performance.

We have established a theoretical convergence which guarantees that the solutions generated by PETRELS-ADMM will converge to a stationary point asymptotically. The simulation results have suggested that our algorithm is more effective than the state-of-the-art algorithms for robust subspace tracking and robust matrix completion. The effectiveness of PETRELS-ADMM was also verified for the problem of video background-foreground separation.

Appendix

Proof of Lemma 1

Follow the line as in previous convergence analysis of ADMM algorithms [START_REF] Li | Global convergence of splitting methods for nonconvex composite optimization[END_REF][START_REF] Wang | Global convergence of ADMM in nonconvex nonsmooth optimization[END_REF], we can derive the proof of Lemma 1 as follows 3.7.1.1 Proof of Proposition (P-1)

The minimizer u k+1 defined in (3.15) satisfies

L s k , u k+1 , r k , w k , e k ≤ L(s k , u k , r k , w k , e k ) -c u u k -u k+1 2 2 . (P-1)
At the k-th iteration, the u-update in fact minimizes the objective function in (3.14), as

u k+1 = argmin u L u,k (u, .) = 1 + ρ 1 2 ∥u∥ 2 2 -P t x t -U t-1 w -ρ 1 s k -r k ⊤ u . (3.69)
The function L u,k (u, .) is strongly convex with a positive constant (1 + ρ 1 ), i.e., the Hessian of

L u,k (u, .) is given by ∇ 2 L u,k (u, .) = (1 + ρ 1 )I. Since u k+1 = argmin u L u,k (u, .
), we have the fact

L u,k (u k+1 , .) ≤ L u,k (u k , .
). Therefore, we obtain the following inequality The minimizer s k+1 defined in (3.18) satisfies

L u,k u k , . -L u,k u k+1 , . ≥ 1 + ρ 1 2 u k+1 -u k 2 2 , ( 3
L s k+1 , u k+1 , r k , w k , e k ≤ L s k , u k+1 , r k , w k , e k -c s s k -s k+1 2 2 . (P-2)
At the k-th iteration, the variable s is updated by minimizing the objective function L s,k (s, .) in Eq. (3.15), as

s k+1 = argmin s L s,k (s, .) = ρ ∥s∥ 1 + ρ 1 2 s -u k+1 + r k 2 2 . (3.71)
We exploit that if given u k+1 and r k , then both functions of the ℓ 1 -norm ∥s∥ 1 and ℓ 2 -norm

s -(u k+1 + r k ) 2 2
are convex, so the L s,k (s, .) w.r.t. s is also convex. It is therefore that for any s k , s k+1 ∈ S, we always have

L s,k s k , . ≥ L s,k s k+1 , . + s k -s k+1 , ∇L s,k s k+1 , . + 1 2 s k+1 -s k 2 2 , (3.72)
thanks to the Proposition 3.

Since s k+1 = argmin s L s,k (s, .), the first derivative ∇L s,k (s k+1 , .) = 0 and hence

L s,k s k , . ≥ L s,k s k+1 , . . (3.73)
In other word, there always exists a nonnegative number c s ≥ 0 such that

L s,k s k , . ≥ L s,k s k+1 , . + 1 2 s k+1 -s k 2 2 . (3.74)
As a result, we have

K k=1 1 2 s k+1 -s k 2 2 ≤ K i=1 L s,k s k , . -L s,k s k+1 , . = L s,k s 1 , . -L s,k s K+1 , . . (3.75) Let K → ∞, we then have ∞ k=1 ∥s k+1 -s k ∥ 2 2 < ∞.
It ends the proof of (P-2) and the second part of Lemma 1.

Proof of Proposition (P-3)

The minimizer r k+1 defined in (3.16) satisfies

L s k+1 , u k+1 , r k+1 , w k , e k ≤ L s k+1 , u k+1 , r k , w k , e k -c r r k -r k+1 2
2 .

(P-3) 3.7. APPENDIX Follow the r-update in Eq. (3.16), it is easy to verify that

L s k+1 , u k+1 , r k+1 , w k , e k = ρ 1 r k + s k+1 -u k+1 ⊤ u k+1 -s k+1 + A = ρ 1 (r k ) ⊤ u k+1 -s k+1 -ρ 1 u k+1 -s k+1 2 2 + A = L s k+1 , u k+1 , r k , w k , e k -ρ 1 r k+1 -r k 2 2 , (3.76)
where A = g(s k+1 ) + h(u k+1 ) + ρ 1 2 ∥u k+1 -s k+1 ∥. It implies the proposition (P-3).

Proof of Proposition (P-4)

The minimizer w k+1 defined in (3.27) satisfies

L s k+1 , u k+1 , r k+1 , w k+1 , e k ≤ L s k+1 , u k+1 , r k+1 , w k , e k -c w w k -w k+1 2 2 .
(P-4)

Denote z = P t (U t w + s k+1 -x t ). In fact, the w-update minimizes the smooth version of the objective function (3.23), as follows

L z,k (z, .) = n i=1 z(i) 2 + 1 1/2 -1 + ρ 2 2 z(i) -e k (i) 2 + 1) 1/2 -1 . (3.77)
The first two derivatives of L z,k (z, .) are given by

∇L z,k (z, .) = z(1)(z(1) 2 + 1) -1/2 , . . . , z(n)(z(n) 2 + 1) -1/2 ⊤ (3.78) + ρ 2 (z(1) -e k (1))((z(1) -e k (1)) 2 + 1) -1/2 , . . . , (z(n) -e k (n))((z(1) -e k (1)) 2 + 1) -1/2 ⊤ ,
and

∇ 2 L z,k (z, .) = diag (z(1) 2 + 1) -3/2 , . . . , (z(n) 2 + 1) -3/2 + ρ 2 diag ((z(1) -e k (1)) 2 + 1) -3/2 , . . . , (z(n) -e k (n)) 2 + 1) -3/2 . (3.79)
The Hessian matrix

∇ 2 L z,k (z, .) then satisfies ρ 2 I < ∇ 2 L z,k (z, .) ≤ (ρ 2 + 1)I. It is therefore that L z,k (w, .
) is strongly convex and Lipschitz continuous. In other word, it implies that

L s k+1 , u k+1 , r k+1 , w k , e k -L s k+1 , u k+1 , r k+1 , w k+1 , e k > ρ 2 2 w k -w k+1 2 2 . (3.80)
which results in the Proposition (P-4), thanks to Proposition 19.

3.7. APPENDIX

Proof of Proposition (P-5)

The minimizer e k+1 defined in (3.29) satisfies L s k+1 , u k+1 , r k+1 , w k+1 , e k+1 ≤ L s k+1 , u k+1 , r k+1 , w k+1 , e k -c e e ke k+1 2 2 .

(P-5)

Similarly, we also have L e,k (e, .) is strongly convex, i.e.,

∇ 2 L e,k (e, .) = ρ 2 diag (z k (1) -e(1)) 2 + 1 -3/2 , . . . , z k (n) -e(n)) 2 + 1 -3/2 . (3.81)
Therefore we have

L e,k e k , . -L e,k e k+1 , . ≥ ρ 2 2 e k+1 -e k 2 2 . (3.82)
It ends the proof.

Proof of Proposition 2

To prove that g t (U) is strongly convex, we state the following facts: g t (U) is continuous and differentiable; its second derivative is a positive semi-definite matrix (i.e., ∇ 2 U g t (U) ⪰ mI); and the domain of g t (U) is convex. In order to satisfy the Lipschitz condition, we show that the first derivative of g t (U) is bounded.

Stage I: Prove that g t is a strong convex function

We show that there exists a positive number m such that

g t (U t+1 ) -g t (U t ) ≥ m 1 ∥U t+1 -U t ∥ 2 F . (3.83)
In particular, we state the two claims as follows:

(C-1): g t (U) is continuous and differentiable.

Proof. Given two variables A, B ∈ U such that ∥A -B∥ 2 F < γ for some positive constant γ. It is easy to verify that there exists a positive number θ such that |g t (A) -g t (B)| < θ.

APPENDIX

Thanks to the triangle inequality, we have the following inequality:

g t (A) -g t (B) = 1 t t i=1 β t-i ∥P i (Aw i + s i -x i )∥ 2 2 - t i=1 β t-i ∥P i (Bw i + s i -x i )∥ 2 2 ≤ 1 t t i=1 β t-i ∥P i (A -B)w i ∥ 2 2 ≤ 1 t t i=1 β t-i ∥P i (A -B)∥ 2 F ∥w i ∥ 2 2 ≤ 1 t t i=1 β t-i ∥A -B∥ 2 F ∥w i ∥ 2 2 = γ t t i=1 β t-i ∥w i ∥ 2 2 = θ, (3.84)
It is therefore that the set of functions {g t (U)} ∞ t=1 is equicontinuous on U.

Furthermore, for any U * , H ∈ U, we show that the following limit exists:

lim a→0 g t (U * + aH) -g t (U * ) a = lim a→0 1 ta t i=1 β t-i P i (U * + aH)w i + s i -x i 2 2 -P i U * w i + s i -x i 2 2 . (3.85)
Specifically, let us denote y i = P i (U * w i + s i -x i ), the limit can be written as follows:

lim a→0 g t (U * + aH) -g t (U * ) a = lim a→0 1 ta t i=1 β t-i ∥y i -aP i Hw i ∥ 2 2 -∥y i ∥ 2 2 = lim a→0 1 ta t i=1 β t-i ∥aP i Hw i ∥ 2 2 -2a⟨u i , P i Hw i ⟩ = -2 t t i=1 β t-i ⟨y i , P i Hw i ⟩ < ∞. (3.86)
As a result, the function g t (U) is differentiable and its first derivative ∇ U g t (U) can be given by

∇ U g t (U) = 2 t t i=1 β t-i P i (Uw i + s i -x i )w ⊤ i . (3.87)
In the similar way, it is easy to verify that ∇ U g t (U) is also continuous and the second derivative ∇ 2 U g t (U) is given by

∇ 2 U g t (U) = 2 t t i=1 β t-i P i w i w ⊤ i . (3.88) 3.7. APPENDIX (C-2): The second derivative ∇ 2 U g t (U
) is a positive-define matrix. For all x ∈ R p×1 , we have

x ⊤ ∇ 2 U g t (U)x = 2 t t i=1 β t-i P i (w ⊤ i x) ⊤ (w ⊤ i x) = 2 t t i=1 β t-i P i (w ⊤ i x) 2 > 0, ∀β, t > 0. (3.89)
It implies that there always exist a positive constant m such that ∇ 2 U g t (U) ≥ mI.

It follows to the claims (C-1), (C-2) and the assumptions showing that the domain of

g t (U) is a convex set that g t (U t ) is strongly convex [132, Section 3.1.4].
Stage II: Prove that g t is a Lipschitz function

g t (U t+1 ) -g t (U t ) ≤ m 2 U t+1 -U t F . (3.90) Let us denote d t (U) = g t (U) -g t+1 (U). Since U t = argmin U∈U g t (U), we exploit that g t+1 (U t+1 ) ≤ g t+1 (U t ) and hence g t (U t+1 ) -g t (U t ) = g t (U t+1 ) -g t+1 (U t ) + g t+1 (U t ) -g t (U t ) ≤ g t (U t+1 ) -g t+1 (U t+1 ) dt(U t+1 ) -g t (U t ) -g t+1 (U t ) dt(Ut) . (3.91)
The first derivative of d t (U) = g t (U) -g t+1 (U) is given by

∇ U d t (U) = ∇ U g t (U) -∇ U g t+1 (U) = 1 t t i=1 β t-i P i (Uw i + s i -x i )w ⊤ i - 1 t + 1 t+1 i=1 β t+1-i P i (Uw i + s i -x i )w ⊤ i . (3.92) Let A t = t i=1 β t-i P i Uw i w ⊤ i and B t = t i=1 β t-i P i (s i -x i ), we can rewrite ∇ U d t (U) as ∇ U d t (U) = A t t - A t+1 t + 1 + B t t - B t+1 t + 1 . (3.93)
Under the assumptions in Section 3.2.2, the subspace U, outlier {s t }, signal {x t } and coefficients {w t } are bounded, then both A t and B t are bounded. It is therefore that

∥∇ U d t (U)∥ F ≤ A t t - A t+1 t + 1 F + B t t - B t+1 t + 1 F ≤ m 2 = O(1/t). (3.94) 3.7. APPENDIX Therefore d t (U) is Lipschiz with the constant m 2 , |d t (U t+1 ) -d t (U t )| ∥U t+1 -U t ∥ F ≤ m 2 , hence |g t (U t+1 ) -g t (U t )| ∥U t+1 -U t ∥ F ≤ m 2 . (3.95)
This ends the proof.

Proof of Lemma 2

We prove that our update rule is an approximate interpretation of Newton's method. Since the objective function g t is strongly convex with respect to the variable U, our algorithm can guarantee that the solution converges to the stationary point of the problem.

In order to estimate subspace, at each time instant t, we optimize the following minimization

u m = argmin u m ∈R r×1 ft (u m ) = t i=1 β t-i P i (m, m) x re i (m) -w ⊤ i u m 2 + α 2t ∥u m ∥ 2 2 . (3.96)
The first derivative of the objective function ft (u m ) can be determined by

∇ ft (u m t-1 ) = -2 t i=1 β t-i P i (m, m) x re i (m) -w ⊤ i u m t-1 w ⊤ i + α t u m t-1 = ∇ ft-1 (u m t-1 ) -2P t (m, m) x re t (m) -w ⊤ t u m t-1 w ⊤ t + α t u m t-1 -u m t-2 . (3.97) Since u m t-1 = argmin u m
ft-1 (u m ), the derivative ∇ ft-1 (u m t-1 ) = 0 and the Hessian at u m t-1 is then given by

H ft (u m t-1 ) = ∇ 2 ft (u m t-1 ) = 2 t i=1 β t-i P i (m, m)w i w ⊤ i + α t I. (3.98)
Thanks to Newton's method [START_REF] Boyd | Convex Optimization[END_REF], a rule for subspace update can be obtained as

u m t = u m t-1 -η t H ft (u m t-1 ) -1 ∇ ft (u m t-1 ). (3.99) Let us denote R m t = t i=1 β t-i P t (m, m)w i w ⊤ i + α 1 2t -βt 2(t-1) I, we have H ft (u m t-1 ) = 2R m t + α β t 2(t -1) - 1 2t I. (3.100)
As a result, we can derive the inverse Hessian matrix easily as follows

H ft (u m t-1 ) -1 = 1 2 R m t -1 O(1/t) 2 R m t -1 + I -1 . (3.101) 3.7. APPENDIX When t is large enough, the term O(1/t) 2 (R m t ) -1 + I -1 ≈ I + O 1 t .
It is therefore that the step size can be approximated by

H ft (u m t-1 ) -1 ∇ ft (u m t-1 ) = -P t (m, m) x re t (m) -w ⊤ t u m t-1 R m t -1 w t + O 1/t . (3.102)
It implies that u m t can be updated by the following recursive update rule

u m t = u m t-1 + η t P t (m, m) x re t (m) -w ⊤ t u m t-1 R m t -1 w t , (3.103)
which is already defined in Eq. (3.14). In other word, the u m t generated by our algorithm can converge to the stationary point of ft (u m ).

Note that, the properties of the objective functions and assumptions we made in Section 3.2.2 can guarantee the method will converge in practice. In particular, the objective functions gt (U) as well as ft (u) and their first derivatives are continuously differentiable which can avoid derivative issues in Newton's method. In addition, the starting points in our algorithm are always chosen at random. Further, since the objective functions {g t (U)} ∞ t=1 are always positive, PETRELS-ADMM can ignore the cases when their roots approach to zero asymptotically. To sum up, the solution U t generated by PETRELS-ADMM will converge to the stationary point of the function gt (U).

The second part of the Lemma 3.7.3 can be easy to verify. Since g t (U t ) is strongly convex and Lipschitz function as proved in Proposition 2, we have the following inequality

m 1 ∥U t+1 -U t ∥ 2 F ≤ g t (U t+1 ) -g t (U t ) ≤ m 2 ∥U t+1 -U t ∥ F ⇔ ∥U t+1 -U t ∥ F ∥U t+1 -U t ∥ F - m 2 m 1 ≤ 0 ⇔ ∥U t -U t+1 ∥ F ≤ m 2 m 1 . (3.104)
Note that the positive number m 2 = O(1/t) is already given in the Appendix 3.7.2, so it ends the proof .

Proof of Lemma 3

Inspired of the result of convergence analysis for online sparse coding framework in [120, Proposition 2], we derive the convergence of g t (U t ) in the similar way. In particular, we first denote the nonnegative stochastic process {u t } as follows

u t ∆ = g t (U t ) ≥ 0, (3.105)
3.7. APPENDIX and then prove that it is a quasi-martingale, i.e., we have to prove the sum of the positive

difference of {u t } ∞ t=1 is bounded, ∞ t=1 E[u t+1 -u t ] < +∞ a.s. (3.106)
We can express g t+1 (U t ) with respect to g t (U t ) as follows

g t+1 (U t ) = 1 t + 1 t+1 i=1 β t+1-i ∥P i (U t w i + s i -x i )∥ 2 2 + ρ ∥s i ∥ 1 = β t + 1 t i=1 β t-i ∥P i (U t w i + s i -x i )∥ 2 2 + ρ ∥s i ∥ 1 + 1 t + 1 ∥P t+1 U t + s t+1 -x t+1 ∥ 2 2 + ρ ∥s t+1 ∥ 1 = βt t + 1 g t (U t ) + 1 t + 1 ℓ(U t , P t+1 , x t+1 ). (3.107) Since U t+1 = argmin U g t+1 (U), we have the fact g t+1 (U t+1 ) -g t+1 (U t ) ≤ 0, f t (U t ) ≤ g t (U t ),
and hence

u t+1 -u t = g t+1 (U t+1 ) -g t (U t ) = g t+1 (U t+1 ) -g t+1 (U t ) ≤0 + g t+1 (U t ) -g t (U t ) ≤ g t+1 (U t ) -g t (U t ) = 1 t + 1 ℓ(U t , P t+1 , x t+1 ) - t(1 -β) + 1 t + 1 g t (U t ). (3.108)
It is therefore that

E[u t+1 -u t ] ≤ E ℓ(U t , P t+1 , x t+1 ) -(t(1 -β) + 1)g t (U t ) t + 1 ≤ E ℓ(U t , P t+1 , x t+1 ) -g t (U t ) t + 1 ≤ E ℓ(U t , P t+1 , x t+1 ) -f t (U t ) t + 1 = E f (U t ) -f t (U t ) t + 1 = E √ t f (U t ) -f t (U t ) E[Gt(Ut)] 1 √ t(t + 1) at , (3.109) because of f t (U t ) ≤ g t (U t ) and E[ℓ(U t , P t+1 , x t )] = f (U t ).
In parallel, we exploit that

G t (U t ) = √ t(f (U t ) -f t (U t ))
is the scaled and centered version of the empirical measure, which converges in distribution to a normal random variable, thanks to the center limit theorem. Hence

E[ √ t(f (U t ) -f t (U t ))
] is bounded with a constant α. Then, the sum of the positive difference . We use the Cauchy-MacLaurin integral test [START_REF] Knopp | Theory and Application of Infinite Series[END_REF] for convergence, as

of u t becomes ∞ t=1 E[u t+1 -u t ] < ∞ t=1 α √ t(t + 1) . ( 3
+∞ t=1 α √ t(t + 1) dt = ∞ x=1 α (x 2 + 1) dx = αarctan(x)| +∞ 1 = α arctan(∞) -arctan(1) < ∞. (3.111)
In other words, since the sum of

a t convergences, hence ∞ t=1 E[u t+1 -u t ] < ∞.
We complete the proof.

Proof of Lemma 4

We investigate the convergence of a surrogate sequence

(g t (U t ) -f t (U t )) 1
t+1 as follows

g t (U t ) -f t (U t ) t + 1 = u t -u t+1 + g t+1 (U t+1 ) -g t+1 (U t ) ≤0 + t(β -1) t + 1 g t (U t ) ≤0 + ℓ(U t , P t+1 , x t+1 ) -f t (U t ) t + 1 ≤ u t -u t+1 (S-1) + ℓ(U t , P t+1 , x t+1 ) -f t (U t ) t + 1 (S-2) , (3.112)
because of u t = g t (U t ) and λ ≤ 1. Note that, (S-1) -(S-2) converge almost surely:

• The sequence E[u t -u t+1 ] converges almost surely as proved in Lemma 3.

• The sequence (S-2) also converges, thanks to the fact E ℓ(U t , P t+1 , x t+1 ) = f (U t ) and the convergence of

E f (Ut)-ft(Ut) t+1
as mentioned in the appendix 3.7.4.

It is therefore that the sequence

(g t (U t ) -f t (U t )) 1 t+1 converges almost surely, i.e., ∞ t=0 g t (U t ) -f t (U t ) 1 t + 1 < ∞. (3.113)
On the other hand, the real sequence

{ 1 t+1 } diverges, ∞ t=0 1 t+1 = ∞. It implies that g t (U t ) - f t (U t ) convergences, thanks to the Proposition 24.

Technical Propositions

Here, we provide the following propositions which help us to derive several important results in our proofs.

APPENDIX Proposition 3 ( [134]

). The function f is strongly convex if and only if for all u, v ∈ dom(f ) we always have

f (v) -f (u) - 1 2 ∥v -u∥ 2 2 ≥ ⟨v -u, θ⟩, ∀θ ∈ ∂f (u).
Proposition 4 ( [START_REF] Boyd | Convex Optimization[END_REF] ). The function f is m-strongly convex, with a constant m if and only if for all u, v ∈ dom(f ) we always have

f (v) -f (u) ≥ m 2 ∥v -u∥ 2 2 .
Proposition 5 ( [132] ). Every norm on R n is convex and the sum of convex functions is convex.

Proposition 6 ( [135]

). The Huber penalty function replaces the ℓ 1 -norm ∥x∥ 1 , x ∈ R n is given by the sum n i=1 f Hub µ (x(i)), where

f Hub µ x(i) =    x(i) 2 2µ , |x(i)| ≤ µ, |x(i)| -µ/2, |x| > µ.
There exists a smooth version of the Huber function f Hub µ , which has derivatives of all degrees,

ψ µ (x) = n i=1 (x(i) 2 + µ 2 ) 1/2 -µ .
and the first derivative of the pseudo-Huber function ψ µ is defined by

∇ψ µ (x) = x(1) x(1) 2 + µ 2 -1/2 , . . . , x(n) x(n) 2 + µ 2 -1/2 ⊤ . Proposition 7 ( [136, Proposition 1.2.4] ). Let {a t } ∞ t=1 and {b t } ∞ t=1 be two nonnegative sequences such that ∞ i=1 a i = ∞ and ∞ i=1 a i b i < ∞, |b t+1 -b t | < Ka t with some constant K, then lim t→∞ b t = 0 or ∞ i=1 b i < ∞.

Introduction

Subspace tracking (ST) is an essential and fundamental problem in signal processing with various applications to sensor array processing, wireless communication, and image/video processing, to name a few [START_REF] Delmas | Subspace tracking for signal processing[END_REF]. It corresponds to the problem of tracking a low-rank subspace that can represent data streams. Most of subspace tracking methods are designed to estimate the underlying subspace from the sample covariance matrix (SCM). We refer the reader to [START_REF] Delmas | Subspace tracking for signal processing[END_REF][START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF]26] for good surveys on standard and robust ST algorithms.

Recently, many rigorous evidences and theoretical results in random matrix theory (e.g. [START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF][START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Vershynin | How close is the sample covariance matrix to the actual covariance matrix?[END_REF]) indicated that the SCM is not a good estimator of the actual covariance matrix in highdimension, low-sample-size (HDLSS) contexts where datasets are massive in both dimension n and sample size T , and typically n/T → c ∈ (0, ∞]. In most online applications, this regime is indeed more realistic and relevant than the classical one where n is fixed and T → ∞. It is mainly due to the time variation of (big) data streams in nonstationary environments where the underlying data distribution changes with time. 1 Accordingly, the data covariance matrix and the principal subspace are time varying too, and thus, the "effective" window length which defines actual data samples under processing is limited. Meanwhile, modern data streams are originally associated with high dimensionality [2]. This leads to the case in which the data dimension n is comparable or even larger than the actual number of snapshots under consideration T .

Without further structural knowledge about the data, subspace tracking algorithms turn out to be inconsistent in such a regime. Interestingly, the consistency of covariance estimation can be guaranteed under suitably structured sparsity regularizations [START_REF] Karoui | Operator norm consistent estimation of large-dimensional sparse covariance matrices[END_REF][START_REF] Lam | Sparsistency and rates of convergence in large covariance matrix estimation[END_REF][START_REF] Johnstone | On consistency and sparsity for principal components analysis in high dimensions[END_REF][START_REF] Bickel | Covariance regularization by thresholding[END_REF][START_REF] Shen | Consistency of sparse PCA in high dimension, low sample size contexts[END_REF]. Therefore, sparse subspace estimation and tracking have recently gained much attention in the signal processing community. In the literature, several good methods have been proposed for sparse subspace estimation, see [START_REF] Cai | Sparse PCA: Optimal rates and adaptive estimation[END_REF][START_REF] Arash | High-dimensional analysis of semidefinite relaxations for sparse principal components[END_REF][START_REF] Journée | Generalized power method for sparse principal component analysis[END_REF][START_REF] Ma | Sparse principal component analysis and iterative thresholding[END_REF] for examples and [START_REF] Zou | A selective overview of sparse principal component analysis[END_REF][START_REF] Cai | Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation[END_REF] for comprehensive surveys. However, in 4.1. INTRODUCTION an adaptive (online) setting, there have been only few studies on sparse subspace tracking (SST) so far.

Related Works

As mentioned before, some online algorithms have been introduced for sparse subspace tracking [26]. A few of them are based on a two-stage approach in which one first utilizes a standard ST algorithm to estimate the underlying subspace and then seek a sparse basis of the estimation under some sparsity criteria. Particularly in [START_REF] Lassami | Low cost subspace tracking algorithms for sparse systems[END_REF][START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF][START_REF] Lassami | Fast sparse subspace tracking algorithm based on shear and givens rotations[END_REF], several variants of OPAST and FAPI were proposed to track the sparse principal subspace. Another good approach is to regularize the objective function that aims at accounting for the sparse basis. In [START_REF] Yang | Fast STAP method based on PAST with sparse constraint for airborne phased array radar[END_REF], the authors modified the objective function of PAST by adding a ℓ 1 -norm regularization term on the subspace matrix and then proposed a new robust variant of PAST called ℓ 1 -PAST to optimize it. Similar to ℓ 1 -PAST, the authors in [START_REF] Xiao | Online sparse and orthogonal subspace estimation from partial information[END_REF] also introduced another adaptive algorithm using ℓ 1 -norm minimization called SPCAur for sparse subspace tracking. SPCAur adopts the stochastic gradient descent on Grassmann manifolds and it is capable of tracking the underlying sparse subspace from incomplete observations. In [START_REF] Giampouras | Online sparse and low-rank subspace learning from incomplete data: A Bayesian view[END_REF], a Bayesian-based algorithm called OVBSL was proposed to deal with the sparsity constraint on the subspace matrix. An advantage of OVBSL is that it is fully automated, i.e., no finetuning parameter is required. However, these algorithms are only effective in the classical regime where the sample size is much larger than the dimension, i.e., n/T → 0 asymptotically.

Through the lens of machine learning and statistics, SST is generally referred to as the problem of online sparse PCA which often emphasizes the leading eigenvectors. In [START_REF] Wang | Online learning for sparse PCA in high dimensions: Exact dynamics and phase transitions[END_REF], the authors proposed an extended version of the Oja algorithm for online sparse PCA, namely OIST. Its convergence, steady-state, and phase transition were also derived to investigate the use of OIST in high dimensions. OIST is, however, designed only for rank-1 sparse subspaces. In [START_REF] Yang | Streaming Sparse Principal Component Analysis[END_REF], another online sparse PCA algorithm (SSPCA) was proposed and could deal with rank-r subspaces.

Specifically, this algorithm uses a simple row truncation operator, which sets rows whose scores are smaller than a threshold to zero, for tracking the sparse principal subspace over time. However, this truncation operator is only designed for subspaces with a row-sparse support (i.e. all eigenvectors must share the same sparsity patterns) which may not always meet in practice. Indeed, it turns out to be ineffective for a sparse subspace with another support (e.g. elementwise sparsity). Its performance in terms of estimation accuracy is typically lower than other SST algorithms, see Fig. It is worth noting that algorithms in [START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF][START_REF] Lassami | Fast sparse subspace tracking algorithm based on shear and givens rotations[END_REF], OIST [START_REF] Wang | Online learning for sparse PCA in high dimensions: Exact dynamics and phase transitions[END_REF], and SSPCA [START_REF] Yang | Streaming Sparse Principal Component Analysis[END_REF] can be viewed as online variants of a classical method for principal subspace estimation, namely power iteration (PI). In the literature, there exist other power-based subspace trackers and they can be broadly categorized into the following classes: Oja-types [START_REF] Abed-Meraim | Orthogonal Oja algorithm[END_REF][START_REF] Allen-Zhu | First efficient convergence for streaming k-PCA: A global, gap-free, and near-optimal rate[END_REF], Natural Power (NP)-types [START_REF] Hua | A new look at the power method for fast subspace tracking[END_REF][START_REF] Abed-Meraim | On a class of orthonormal algorithms for principal and minor subspace tracking[END_REF],

4.1. INTRODUCTION Data Projection Method (DPM)-types [START_REF] Doukopoulos | Fast and stable subspace tracking[END_REF][START_REF] Wang | A novel orthonormalization matrix based fast and stable DPM algorithm for principal and minor subspace tracking[END_REF], and Approximated PI (API)-types [START_REF] Badeau | Fast approximated power iteration subspace tracking[END_REF][START_REF] Wu | An improved adaptive subspace tracking algorithm based on approximated power iteration[END_REF].

Specifically, all of them are designed for tracking the principal subspace of the SCM which is, however, not a good estimator of the true data covariance matrix in high dimensions. Accordingly, they turn out to be inconsistent estimators in the HDLSS regime.

In parallel, recent years have also witnessed considerable research advances on robust ST (RST) which aims to track the underlying subspace in the presence of data corruption [START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF]26,[START_REF] Vaswani | Static and dynamic robust PCA and matrix completion: A review[END_REF].

For example, several RST algorithms were developed to handle sparse outliers, such as Grassmannian Robust Adaptive Subspace Tracking Algorithm (GRASTA) [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], Parallel Subspace Estimation and Tracking by Recursive Least Squares (PETRELS)-types [25,[START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF], and Recursive

Projected Compressive Sensing (ReProCS)-types [START_REF] Narayanamurthy | Provable dynamic robust PCA or robust subspace tracking[END_REF][START_REF] Narayanamurthy | Provable subspace tracking from missing data and matrix completion[END_REF]. To deal with impulsive noises, three potential approaches are robust statistics [START_REF] Zhang | A robust correntropy based subspace tracking algorithm in impulsive noise environments[END_REF][START_REF] Chan | A robust PAST algorithm for subspace tracking in impulsive noise[END_REF], adaptive Kalman filtering [START_REF] Liao | A new robust Kalman filter-based subspace tracking algorithm in an impulsive noise environment[END_REF][START_REF] Chan | A new adaptive Kalman filter-based subspace tracking algorithm and its application to DOA estimation[END_REF], and weighted RLS [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF][START_REF] Nguyen | Robust subspace tracking algorithms using fast adaptive Mahalanobis distance[END_REF]. Very recently, α-divergence was specifically exploited to bolster the tracking ability of the well-known PAST and FAPI trackers in noisy and contaminated environments [START_REF] Rekavandi | TRPAST: A tunable and robust projection approximation subspace tracking method[END_REF][START_REF] Thanh | Robust subspace tracking with contamination via α-divergence[END_REF].

However, none of them is designed for subspace tracking in the HDLSS context.

Contribution and Significance

In this chapter, we introduce a new provable adaptive algorithm called OPIT (OPIT stands for Online Power Iteration via Thresholding) for sparse subspace tracking. OPIT takes both advantages of power iteration and thresholding methods, and hence offers several appealing features over the state-of-the-art SST/online sparse PCA algorithms.

First, OPIT belongs to the class of power methods, and thus its convergence rate is highly competitive compared to other SST algorithms, especially in the high SNR regime. Unlike the two SST algorithms based on power methods (i.e. OIST and SSPCA), OPIT utilizes old observations efficiently in a recursive way and still operates with linear space complexity. Accordingly, OPIT could obtain not only a faster convergence rate but also a better subspace estimation accuracy than OIST and SSPCA. Compared to OIST which is limited to tracking rank-1 sparse subspaces, OPIT has the capability of tracking rank-r subspaces over time. Compared to SSPCA which is useful for only subspaces with row-sparse supports, OPIT offers an effective subspace tracker which can deal with more generalized sparsity supports than SSPCA, thanks to a new thresholding operator to deal with subspace sparsity. In particular, we propose to apply columnbased thresholding instead of row-based thresholding as in SSPCA. With this operator, OPIT has a great potential for handling several sparsity supports such as row-sparse, elementwise-sparse, and local region-sparse.

Different from the existing two-stage SST algorithms, OPIT has ability to track the sparse principal subspace with high accuracy in both the classical regime and the HDLSS regime.

Theoretically, the subspaces derived from the two-stage algorithms are identical to those obtained 4.2. PROBLEM FORMULATION by the corresponding standard ST algorithms (e.g. OPAST and FAPI) used in their first stage.

It is due to the fact that the subspace spanned by a full rank matrix remains unchanged after any rotation. Accordingly, they still suffer the limitation of the SCM in the HDLSS regime. By contrast, our OPIT algorithm aims to track the underlying sparse subspace from a thresholded SCM. Simulation results indicate that OPIT provides a much better subspace estimation accuracy than the two-stage SST algorithms in high dimensions. More importantly, as indicated later in our theoretical analysis, the convergence of OPIT with the thresholding operation can be guaranteed under certain conditions.

In addition, OPIT is flexible and very adaptable for different scenarios. In particular, we can adjust its procedure for dealing with multiple incoming data streams. This feature is useful for application areas wherein block processing is required, i.e., a block of data samples is processed and analysed at one time. Next, it is easy to introduce regularization parameters into OPIT in order to regularize its performance in non-standard environments. Specifically, we can use a forgetting factor to discount the impact of distant observations as well as facilitate the tracking ability of OPIT in dynamic environments. Moreover, we can recast its update rule into a columnwise update. Thanks to the deflation transformation, we particularly derive a fast variant of OPIT called OPITd with lower complexity of both computation and memory storage. This variant is fast and useful for tracking high-dimension and large-scale data streams residing in a low-dimensional space. Last but not least, OPIT belongs to the class of provable subspace tracking algorithms in which its convergence is guaranteed. Under certain conditions, OPIT can achieve an ϵ-relative-error approximation with high probability when the number of observations is large enough.

Organization and Notations

The rest of the chapter is organized as follows. Section 4.2 formulates the SST problem. Section 7.3.2 presents the proposed OPIT algorithm and its variant OPITd while Section 4.4 establishes its convergence analysis. Section 4.5 provides several experiments to demonstrate performance of the proposed algorithms in comparison with the state-of-the-art algorithms. Section 4.6 concludes the chapter.

Problem Formulation

Assume that at time t, we collect a data sample x t ∈ R n×1 satisfying the signal model

x t = ℓ t + n t . (4.1)

PROBLEM FORMULATION

Here, ℓ t ∈ R n×1 is a low-rank signal living in a subspace2 spanned by a sparse matrix A n×r with r < n (i.e. ℓ t = Aw t , where w t ∈ R r×1 is a weight vector) and n t ∈ R n×1 is an additive spatially white noise vector independent of ℓ t . Sparse subspace tracking problem can be stated as follows:

Sparse Subspace Tracking: Given a set of data streams {x t } T t=1 , we aim to estimate a sparse principle subspace A t that compactly represents the span of signals {ℓ t } T t=1 .

Generally, the underlying subspace can be estimated from the spectral analysis of the actual covariance matrix

C = E x t x ⊤ t = AE w t w ⊤ t A ⊤ + E n t n ⊤ t . (4.2)
Without loss of generality, we suppose that C has the form

C = σ 2 x AA ⊤ +σ 2 n I n where E{w t w ⊤ t } = σ 2 x I r and E{n t n ⊤ t } = σ 2 n I n . Applying eigenvalue decomposition (EVD) on C yields C EVD = UΛU ⊤ = U s U n   Λ s 0 0 Λ n     U ⊤ s U ⊤ n   . (4.3)
Here, Λ ∈ R n×n is a diagonal matrix whose diagonal elements are eigenvalues of C sorted in decreasing order and U ∈ R n×n contains the corresponding eigenvectors. Accordingly, U s ∈ R n×r and U n ∈ R n×(n-r) represent the principal subspace and the minor subspace of C, respectively.

The orthogonal projection matrix of the sparse principal subspace is unique (i.e., U s U ⊤ s = AA # ), so A can be obtained as

A = U s Q * with Q * = argmin Q∈R r×r U s Q 0 s.t. Q is full-rank, (4.4)
where ∥.∥ 0 promotes the sparsity on A. In several applications, we often emphasize the principal subspace rather than its specific basis, such as dimensionality reduction [START_REF] Van Der Maaten | Dimensionality reduction: A comparative review[END_REF] and array processing [START_REF] Dung | Generalized minimum noise subspace for array processing[END_REF]. In this work, our main objective is to track the principal (signal) subspace of A while the sparsifying step (4.4) is optional.

Most state-of-the-art SST algorithms estimate the principal subspace of the sample covariance [26]. However, in a high-dimensional regime where n/T ↛ 0 a.s., C T is not a good estimator of C. This limitation in an adaptive scheme is not necessarily due to a data shortage but to the time variation which forces us to use a limited window of time instead of all the data. Particularly, it has been shown that C T is not a consistent estimate of C in the HDLSS regime, e.g. [START_REF] Chaufray | Asymptotic analysis of optimum and suboptimum CDMA downlink MMSE receivers[END_REF][START_REF] Mestre | Modified subspace algorithms for DoA estimation with large arrays[END_REF][START_REF] Cai | Optimal rates of convergence for covariance matrix estimation[END_REF]. As a result, most of SST algorithms are not good in high dimensions, as illustrated in Fig. 4.5.

matrix C T = 1/T T t=1 x t x ⊤ t

PROPOSED METHODS

On the other hand, under certain conditions, it is proved in [START_REF] Karoui | Operator norm consistent estimation of large-dimensional sparse covariance matrices[END_REF][START_REF] Rothman | Generalized thresholding of large covariance matrices[END_REF] that

C -τ (C T ) 2 → 0 a. s. as T → ∞, (4.5)
where τ (.) is an appropriate thresholding operator. Thanks to (4.5), in the next section, we derive a novel adaptive (online) algorithm based on power iteration and thresholding technique that is capable of tracking the sparse principal subspace in both the classical regime and the HDLSS regime.

Proposed Methods

In this section, a novel effective algorithm using thresholding is developed for sparse subspace tracking. This algorithm is dubbed as OPIT which stands for Online Power Iteration via Thresholding. We next derive a fast variant of OPIT called OPITd with lower complexity, thanks to the deflation transformation. Some remarks on OPIT and OPITd are discussed in the following subsection.

OPIT Algorithm

We first recall the main steps of the standard power iteration (PI) method on which we primarily leverage in order to develop our OPIT algorithm, for computing the dominant eigenvectors of C t . At the ℓ-th iteration, PI particularly updates (i) S ℓ ← C t U ℓ-1 and (ii) U ℓ ← QR(S ℓ ) be the Q-factor of QR factorization of S ℓ . PI starts from an initial matrix U 0 ∈ R n×r and returns an orthonormal matrix U L where L is the number of iterations [START_REF] Delmas | Subspace tracking for signal processing[END_REF].

In an adaptive scheme, the iteration step of PI can coincide with the data collection in time.

At time t, the sample covariance matrix C t can be recursively updated by: R t

= R t-1 + x t x ⊤ t and C t = t -1 R t .
As streaming data can vary with time, we propose to use a forgetting factor β (0 < β ≤ 1) to discount the impact of old observations exponentially. The underlying subspace U t is then derived from spectral analysis of R t which is updated continuously by

R t = βR t-1 + x t x ⊤ t . (4.6)
Together with the fact that QR(R t U t-1 ) = QR(C t U t-1 ), we can rewrite the first step of PI as follows

S t = R t U t-1 = βR t-1 U t-1 + x t z ⊤ t , (4.7)
where

z t = U ⊤ t-1 x t . 4.3. PROPOSED METHODS Algorithm 4: OPIT Input: {x i } T i=1 , x i ∈ R n×1 , target rank r, a forgetting factor 0 < β ≤ 1, window of length W ≥ 1, and a thresholding factor k k = ⌊(1 -ω sparse )n⌉ if ω sparse is given, ⌊10r log n⌉ if ω sparse is unknown,
where ω sparse is the sparsity level of the sparse basis.

Initial: U 0 = randn(n, r), S 0,F = 0 n×r , E 0 = 0 r×r Main Program: Procedure for t = 1, 2, . . . , T /W do X t = [x (t-1)W +1 , . . . , x tW ] // Data collection Z t = U ⊤ t-1 X t S t = β (t-1)W tW S t-1 E t-1 + 1 tW X t Z ⊤ t Ŝt = τ (S t , k) // Thresholding U t = QR( Ŝt ) // Promotes orthogonality Ŝt /∥ Ŝt ∥ 2 // Promotes sparsity E t = U ⊤ t-1 U t end for Output: U t ∈ R n×r // Thresholding Ŝt = τ (S t , k) Procedure for i = 1, 2, . . . , r do s i = S t (:, i) Find the set T t ⊂ [1, 2, . . . , n] containing indices of k strongest elements of s i Form Ŝt (:, i) = ŝi , where ŝi (j) = s i (j) if j ∈ T t 0 if j / ∈ T t end for Output: Ŝt ∈ R n×r
Towards a fast subspace estimator, we can utilize the previous subspace as a warm start in the tracking process. Hereby, a key step at each time t is to project U t into the column space of

U t-1 , i.e., U t = U t-1 E t + U t-1,⊥ F t , (4.8)
where

U t-1,⊥ is the orthogonal complement of U t-1 , E t = U ⊤ t-1 U t and F t = U ⊤ t-1
,⊥ U t are coefficient matrices. Specifically, the first term of (4.8) represents the "old" information in U t , while the second one is its distinctive new information. Substituting U t-1 according to (4.8)

PROPOSED METHODS

(one time-step delayed) into (4.7) results in

S t = βS t-1 E t-1 + βR t-1 U t-2,⊥ F t-1 + x t z ⊤ t . (4.9)
The complement of projecting x t into the subspace U t-1 at time t can be given by

y t = I -U t-1 U ⊤ t-1 x t = x t -U t-1 z t . (4.10)
Here, y t is orthogonal to the column space of U t-1 . For short, we denote

∆U t-1 = U t-2,⊥ F t-1 .
Based on (4.10), we obtain another expression of ∆U t-1 as follows

∆U t-1 = y t-1 h ⊤ t-1 where h t-1 = U ⊤ t-1 y t-1 . (4.11)
Under the assumption that the underlying subspace is fixed or slowly varying with time (i.e.,

U t-2 U ⊤ t-2 ≃ U t-1 U ⊤ t-1
), y t-1 is nearly orthogonal to the subspace U t-1 . In other words, angles between y t-1 and columns of U t-1 are very close to π/2, and hence, the norm of h t-1 in (4.11) is very small. Therefore, ∆U t-1 and R t-1 ∆U t-1 are negligible and can be ignored during the tracking process without any major performance degradation. It stems from the fact that the presence of a small perturbation does not really affect the performance of power methods [START_REF] Hardt | The noisy power method: A meta algorithm with applications[END_REF].

Accordingly, a good approximation to (4.9) can be given by

S t ≃ βS t-1 E t-1 + x t z ⊤ t . (4.12)
In this work, the update (4.12) is further followed by an appropriate perturbation G t defined by the following thresholding operation τ (.) as:

Ŝt ∆ = τ (S t , k) = C t U t-1 + G t , (4.13)
where the thresholding factor k can be determined as in Algorithm 4. Here, Ŝt is particularly derived from S t by keeping the k strongest (absolute value) elements in each column of S t and setting the remaining elements to zero. Then, the second step of PI is replaced with

U t =    QR( Ŝt ) if orthonormalization, Ŝt ∥ Ŝt ∥ 2 if normalization. (4.14)
In addition to the nice property (4.5), another main motivation for using the thresholding operation τ (.) stems from the following proposition:

4.3. PROPOSED METHODS Proposition 9. Denote by {λ i } n i=1 the set of singular values of C t in descending order (i.e.

λ i ≥ λ i+1 ). When the perturbation G t satisfies: ∥G t ∥ 2 ≤ ξ(λ r -λ r+1 ) and ∥A ⊤ t G t ∥ 2 ≤ ξ(λ r -λ r+1 ) cos θ(A t , U t-1 ) for some ξ < 1, we obtain tan θ A t , C t U t-1 + G t ≤ γ tan θ A t , U t-1 ,
where 0 < γ < 1 and θ(., .) denotes the canonical angle (the largest principal angle) between two subspaces.

Proof. Its proof follows immediately Lemma 2.2 in [START_REF] Hardt | The noisy power method: A meta algorithm with applications[END_REF].

As a corollary, the estimated U t will get closer to the true subspace A t with time.

The OPIT algorithm introduces the window parameter W . Here, the inclusion of W is useful in some applications where we often collect multiple data samples instead of a single sample at each time t. The main steps of OPIT are summarized in Algorithm 4.

Complexity: For convenience of analysis, we suppose the window length W = 1. Most of the steps in OPIT require a computational complexity of O(nr 2 ) except the thresholding operator which costs O(nr + rk log k) operations. Thus, the overall computational complexity of OPIT is O(max{nr, k log k}r). In terms of memory storage, OPIT does not need to go back past observations but utilizes their information in a recursive way. Hence, the proposed algorithm requires a space of nr elements for saving the estimate U t , while two buffer matrices S t and E t need only nr + r 2 elements in total. In conclusion, the space complexity of OPIT is linear to the data dimension n.

OPIT with Deflation

A low cost subspace tracking algorithm with linear complexity of computation O(nr) is always preferable due to its fast implementation time, especially for real-time applications. 3 Here, we derive a fast variant of OPIT using deflation called OPITd which can achieve such a complexity while preserving the algorithm's accuracy in most cases.

Our main motivation stems from the fact that if we apply the following projection deflation

C t = (I -u 1 u ⊤ 1 )C t (I -u 1 u ⊤ 1 ), (4.15)
where u 1 is the most dominant eigenvector of C t , then the eigenvectors of C t are exactly the

PROPOSED METHODS

Algorithm 5: OPITd -OPIT with Deflation

Input: {x i } T i=1 , x i ∈ R n×1 , target rank r, a forgetting factor 0 < β ≤ 1, and a thresholding factor

k k = ⌊(1 -ω sparse )n⌉ if ω sparse is given, ⌊10r log n⌉ if ω sparse is unknown,
where ω sparse is the sparsity level of the sparse basis.

Initial: U 0 = randn(n, r), S 0 = 0 n×r , e 0 = 1 r×1 . // Denote u t,j = U t (:, j), s t,j = S t (:, j), and e t,j = e t (j).

Main Program:

Procedure for t = 1, 2, . . . , T do for j = 1, 2, . . . , r do

z t,j = u ⊤ t-1,j x t s t,j = β t-1 t e t-1,j s t-1,j + 1 t z t,j x t ŝt,j = τ (s t,j , k) // Thresholding u t,j = ŝt,j /∥ŝ t,j ∥ 2 e t,j = u ⊤ t-1,j u t,j x t = x t -z t,j u t,j
// Deflation end for end for Output: U t ∈ R n×r same as C t with eigenvalues {0, λ 2 , . . . , λ n }. Here, λ i is the i-th strongest eigenvalue of C t . It demonstrates that the deflation (4.15) can eliminate the influence of u 1 (i.e., by setting λ 1 to zero) and switches the second dominant eigenvector up. As a result, once we estimated u 1 by using a specific (online) method, the second dominant eigenvector of C t can be extracted from C t in the same way as to u 1 . Moreover, repeating this procedure r times can result in r leading eigenvectors of C t . Interestingly, in the case even when u 1 is not a true eigenvector of C t , the projection deflation (4.15) still retains desirable properties (e.g. positive semi-definiteness) that may be lost to other deflation transformations [START_REF] Mackey | Deflation methods for sparse PCA[END_REF]. In what follows, we describe the way how to linearize the production of OPIT using the projection deflation (4.15).

To update the j-th column u t,j of U t , for j = 1, 2, . . . , r, we replace the recursive rule (4.12) with

s t,j = β t -1 t e t-1,j s t-1,j + 1 t z t,j x t , with (4.16a) z t,j = u ⊤ t-1,j x t and e t-1,j = u ⊤ t-2,j u t-1,j , (4.16b)
where s t,j , z t,j , and e t-1,j play the same role as S t , z t , and E t in (4.12), respectively. Next, the 4.3. PROPOSED METHODS thresholding operation (4.13) boils down to ŝt,j = τ (s t,j , k). (4.17)

Then, the column u t,j is simply derived from normalizing (4.17) to unit length as u t,j = ŝt,j /∥ŝ t,j ∥ 2 . At the end of the column-wise update, we deflate the component u t,j from x t as x t ← x t -z t,j u t,j for the estimation of the next component u t,j+1 . The main steps of OPITd are summarized in Algorithm 5.

Complexity:

The most expensive computation comes from the thresholding operation τ (s t,j , k)

which requires a cost of O(n + k log k).

Discussions

First, it is worth noting that both OPIT and OPITd cannot enforce orthogonality and sparsity in the estimate at the same time. On the one hand, when we adopt the orthonormalization step using the QR factorization, OPIT ensures orthogonality but lacks sparsity. Although performing the QR step can increase the numerical stability of OPIT, it destroys the sparsity, especially when the target rank r is not too small. In most cases, the Q-factor of the thresholded Ŝt is a dense (orthogonal) matrix. However, when the columns of Ŝt are sufficiently sparse and have mostly non-zero elements in non-overlapping sets in its row support, then Ŝt is almost orthogonal and its Q-factor can be nearly sparse. We particularly meet such a case when data streams are high-dimensional but of very low rank (i.e., r ≪ n) and/or the sparsity level ω sparse is extremely high. In fact, we often emphasize the principal subspace rather than its specific basis in subspace tracking, thus the lack of sparsity of OPIT is not the issue. On the other hand, when the normalization step (e.g. OPITd promotes sparsity but entails non-orthogonality and sub-optimality. Thanks to the projection deflation, OPITd offers a fast column-wise update for tracking the underlying sub-4.3. PROPOSED METHODS space and successes in achieving the sparsity. The deflation has the advantage to estimate the eigenvectors (which is referred to as principal components) while the matrix U t in OPIT can be any basis of the principal subspace (not necessarily the eigenvectors). Accordingly, OPITd has benefits in some applications such as data whitening requiring the eigenvectors. Specifically, the combination of the thresholding operation τ (s t,j , k) and the column normalization results directly in sparse components in the estimate U t at each time t. However, the deflation may cause loss of orthogonality and introduces cumulative errors which can affect the successive estimation of the next component. Accordingly, when the target rank r is not too small compared to the data dimension n, both convergence rate and estimation accuracy of OPITd are less than that of OPIT, see Fig. 4.7(b) for an illustration. In such a situation, we can re-orthonormalize U t after a period of time to remedy the issue at low cost as well as increase the numerical stability of OPITd.

U t = Ŝt /∥ Ŝt ∥ 2 ) is
Next, how to choose the value of k? Ideally, this factor must be a r

× 1 vector [k 1 , k 2 , . . . , k r ]
where k j represents the threshold level for the j-th column A t (:, j). Clearly, the value of k j should be close to the number of non-zero elements in A t (:, j). Without loss of generality, we can assume that sparse patterns in A t are uniformly distributed, i.e., k i ≃ k j ∀i, j. Accordingly, we can set k ≃ k j ≃ ⌊(1 -ω sparse )n⌉ when the prior knowledge of the sparsity level ω sparsethe percentage of non-zero elements in A t -is given. If this information is not available, we can tune this factor through cross-validation or simply chosen in O(log n), e.g. k = ⌊mr log n⌉ where m is a positive number. The former remedy is useful for batch sparse subspace estimation and sparse PCA [START_REF] Yuan | Truncated Power Method for Sparse Eigenvalue Problems[END_REF]. However, it requires a validation set -which we have to pass a number of observations several times -and hence turns out to be inefficient for tracking problems. The latter one is very simple and capable of achieving reasonable performance in practice. It stems from the rigorous evidence in [START_REF] Deshpande | Sparse PCA via covariance thresholding[END_REF][START_REF] Wang | Statistical and computational trade-offs in estimation of sparse principal components[END_REF][START_REF] Krauthgamer | Do semidefinite relaxations solve sparse PCA up to the information limit?[END_REF] that sparse subspace/PCA algorithms can recover the sparse principal components in polynomial time when the expected number of non-zero elements in each component is at most O( T / log n). As indicated later in Section IV, the number of observations T = O(n) can guarantee OPIT's convergence, please see the condition (4.18).

Furthermore, we have log n < T / log n when T = O(n) for a large n, and thus, we can choose the factor k in the logarithmic regime O(log n) to ensure the thresholded matrix is sufficiently sparse. A natural question raised here is whether the tracking ability of OPIT deteriorates or not when the number of selected elements is smaller than the actual number of non-zeros in A t ? (e.g. it might occur due to the low level of sparsity). Fortunately, Proposition 9 also suggests that if the perturbation error caused by the choice of k is small enough, OPIT still results in a good estimate of A t when the number of observation is large enough.

Compared to the state-of-the-art power-based subspace tracking algorithms, OPIT is more elegant, refined, and effective. Particularly upon the arrival of new data x t , many powerbased subspace trackers (e.g., Oja-types, NP-types, and DPM-types) adopt the update rule 4.4. CONVERGENCE ANALYSIS

U t = orthnorm(U t-1 + η t x t z ⊤ t )
where η t is the step size and orthnorm(.) is an orthonormalization procedure [START_REF] Delmas | Subspace tracking for signal processing[END_REF]. Therefore, the inclusion of E t-1 in (4.12) not only makes OPIT different from them, but also greatly bolsters its tracking ability. The matrix E t-1 , which contains cosines of the principal angles between two successive subspaces, plays the role of feedback in the tracking process. Accordingly, it could help improve the adaptation rate and stability of OPIT, especially in nonstationary environments. API-type subspace trackers, on the other hand, exploit the projection approximation U t ≃ U t-1 Θ t where Θ t is nearly orthogonal and very close to an identity matrix [START_REF] Badeau | Fast approximated power iteration subspace tracking[END_REF]. Hereby, they would predict the current tracking performance error and then use it for estimating the true subspace. More specifically, they follow the update rule

U t = U t-1 Θ t + y t g ⊤ t Θ
t where y t is the complement (error) of projecting x t onto U t-1 defined as in (4.10), g t is a gain vector, and Θ t = (I r + ∥y t ∥ 2 g t g ⊤ t ) -1/2 . However, when abrupt changes happen (e.g., due to impulsive noises and outliers or data drift), the error y t would be very large. The state transition matrix Θ t would be very far from ideal that could degrade their subspace estimation accuracy as well as convergence rate, see Section E.1 in our supplementary document for examples. By contrast, OPIT exploits the past tracking performance error (i.e., one time step delayed) caused by itself which is independent of the current error y t . Thus, OPIT is less sensitive to such changes than API-types. Together with the hard-thresholding operator τ (.) in (4.13), OPIT stands out from all the rest. The tracking ability of OPIT is verified by several experiments in Section V where the results indicate that OPIT outperforms completely the-state-of-the-art subspace trackers (including several power-based methods) in both classical and high dimension regimes.

Convergence Analysis

In this section, we provide a convergence analysis for the proposed OPIT algorithm in Algorithm 4 under the assumption that A t = A is unchanged over time and β = 1. 4We make the following assumptions to facilitate our convergence analysis:

(A1) A is chosen in the set U = {U ∈ G n,r , ∥U∥ * ,0 ≤ (1 -ω sparse )n, and ∥U∥ 2 = 1}, where
G n,r denotes the class of n × r well-condition matrices and ∥U∥ * ,0 = max j ∥U(:, j)∥ 0 . Here, the parameter ω sparse represents the sparsity level of A. In addition, A is sparse enough in the sense that the average number of non-zero elements in each column is at most n/ log n.

(A2) Data samples {x t } t≥1 are norm-bounded, i.e., ∥x t ∥ 2 ≤ M < ∞ ∀t. Low-rank signals 4.4. CONVERGENCE ANALYSIS {ℓ t } t≥t are supposed to be deterministic and bounded. Noise vectors {n t } t≥1 are i.i.d. random variables of zero mean and their power is lower than the signal power.

In (A1), the underlying subspace is supposed to be sparse in the sense of column sparsity defined by Vu et al. in [START_REF] Vu | Minimax sparse principal subspace estimation in high dimensions[END_REF]. 5 It is not a strict sparsity constraint as the set U covers several supports such as row-sparse, elementwise-sparse, and local region-sparse. Besides, the unit-norm constraint of (A1) is a very mild condition as we can rescale A by recasting its operator norm into the signal power. The second constraint of (A1) ensures trackers to estimate the sparse subspace with high probability [START_REF] Deshpande | Sparse PCA via covariance thresholding[END_REF]. Meanwhile, (A2) is a common assumption for subspace tracking problems and holds in many situations [25]. Together with (A1), they help prevent the ill-conditioned computation and support the perturbation analysis of QR decomposition due to the thresholding operation.

Given these assumptions, the main theoretical result of OPIT's convergence can be stated by the following theorem: Theorem 3. Suppose that A t = A, β = 1, the true covariance matrix has the form

C = σ 2
x AA ⊤ + σ 2 n I, and two assumptions (A1)-(A2) are met. The initialization matrix U 0 and the number of observed (block) data samples t satisfies the following conditions

t ≥ C log(2/δ) W ϵ 2 √ r + σ 2 n σ 2 x + 2 σ n σ x √ n 2 , (4.18)
max sin θ(A, U 0 ), ϵ ≤ 3 -2 √ 2 r + 2 √ r( √ 2 -1) 1/2 , (4.19)
where ϵ > 0 is a predefined accuracy, C is a universal positive number and 0 < δ ≪ 1 is a predefined error probability. At time t, when U t is generated by OPIT with the orthonormalization step using QR factorization, then

d t ∆ = sin θ(A, U t ) ≤ ϵ, (4.20)
with a probability at least 1 -δ.

Proof Sketch. First, let us denote the QR decomposition of S t by S t = U t,F R t,F where "F" stands for "full" entries. Here, we can express

U t = U t,F W 1 + U t,F ,⊥ W 2 where U t,F ,⊥ ∈ R n×(n-r) is the orthogonal complement of U t,F (i.e., U ⊤ t,F U t,F ,⊥ = 0), W 1 ∈ R r×r and W 2 ∈ R (n-r)×r are coefficient matrices. Specifically, it is easy to obtain that ∥W 1 ∥ 2 = ∥U ⊤ t,F U t ∥ 2 and 4.4. CONVERGENCE ANALYSIS ∥W 2 ∥ 2 = ∥U ⊤ t,F ,⊥ U t ∥ 2 .
Accordingly, we can bound the distance d t = sin θ(A, U t ) as follows:

6 d t = A ⊤ ⊥ U t 2 (i) = A ⊤ ⊥ U t,F W 1 + U t,F ,⊥ W 2 2 (i) ≤ A ⊤ ⊥ U t,F 2 W 1 2 + A ⊤ ⊥ U t,F ,⊥ 2 W 2 2 (ii) ≤ A ⊤ ⊥ U t,F 2 + U ⊤ t,⊥ U t,F 2 . (4.21)
Here, (i) thanks to the standard inequalities ∥M + N∥ 2 ≤ ∥M∥ 2 + ∥N∥ 2 and ∥MN∥ 2 ≤ ∥M∥ 2 ∥N∥ 2 ; and (ii) is due to the following facts:

∥A ⊥ ∥ 2 = ∥U t ∥ 2 = ∥U t,F ,⊥ ∥ 2 = 1, ∥W 1 ∥ 2 ≤ ∥U ⊤ t,F ∥ 2 ∥U t ∥ 2 ≤ 1, ∥A ⊤ ⊥ U t,F ,⊥ ∥ 2 ≤ ∥A ⊤ ⊥ ∥ 2 ∥U t,F ,⊥ ∥ 2 ≤ 1, and ∥U ⊤ t,F ,⊥ U t ∥ 2 = ∥U ⊤ t,⊥ U t,F ∥ 2 .
The two terms of the right hand side of (4.21) can be bounded by Lemma 5 and 6, respectively.

Lemma 5. Let ∆C t = C t -C, we always have

A ⊤ ⊥ U t,F 2 ≤ σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 σ 2 x + σ 2 n 1 -A ⊤ ⊥ U t-1 2 2 -∆C t 2 2 + σ 2 n A ⊤ ⊥ U t-1 2 + ∆C t 2 2 1/2 . (4.22)
Proof. See Appendix A.

Lemma 6. The distance between U t and U t,F is bounded by

U ⊤ t,⊥ U t,F 2 ≤ √ r σ 2 n A ⊤ ⊥ U t-1 2 + ∆C t 2 (σ 2 x + σ 2 n ) 1 -A ⊤ ⊥ U t-1 2 2 -1 + √ r(1 + √ 2) σ 2 n A ⊤ ⊥ U t-1 2 + ∆C t 2 , (4.23)
under the following condition

σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 (σ 2 x + σ 2 n ) 1 -A ⊤ ⊥ U t-1 2 2 ≤ √ 2 -1 √ r -1 + √ 2 . ( 4
∆C t 2 ≤ c δ σ 2 x r tW + 2σ n σ x + σ 2 n n tW , (4.25)
where δ > 0 is a predefined error probability, and c δ = C log(2/δ) with a universal positive number C > 0.

Proof. See Appendix C.

Then, the necessary condition (4.24) for Lemma 6 is particularly satisfied when (4.18) is met and the following inequality holds

max sin θ(A, U 0 ), ϵ ≤ α(r, ρ) 1 -α(r, ρ)
, where (4.26)

α(r, ρ) = (3 -2 √ 2)(σ 2 x + σ 2 n ) 2 r + 2 √ r( √ 2 -1) + 3 -2 √ 2 σ 2 n + r -1 ρσ 2 x 2 , (4.27)
for any positive number ρ in the range (0, r], please see Appendix D for details. Clearly, (

provides a lower bound on α(r, ρ)/(1 -α(r, ρ)).

Accordingly, Lemma 6 is achieved under the two conditions (4.18) and (4.19) while Lemma 1 holds for all t. Now, given Lemma 5, 6, and 7, the distance d t can be bounded by Lemma 6.

Lemma 8. Let d 0 = sin θ(A, U 0 ), ω 0 = max{d 0 , ϵ}, γ > 0 is any positive number satisfying ω 0 ≤ γr 1 -ω 2 0 and ργ < 1. Suppose that ω 0 ≤ √ 2/2, the two conditions (4. [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF] and (4.18) are met, we obtain

d t ≤ rσ 2 n + ρσ 2 x rξ 1 -ω 2 0 max d t-1 , ϵ , where (4.28)
ξ = 0.5 max (1 + γ 2 r 2 )σ 4 n + (1 -ργ) 2 σ 4 x + 2(1 + γ 2 r 2 -ργ)σ 2 n σ 2 x 1/2 , (σ 2 n + σ 2 x )(1 -ϱ)/ √ r , (4.29
)

with ϱ = γ 1 + √ r(1 + √ 2)(rσ 2 n + ρσ 2 x ) σ 2 n + σ 2 x -1
. Furthermore, d t ≤ ϵ also holds when t satisfies the condition (4.18).

Proof. See Appendix D.

4.5. EXPERIMENTS

Experiments

In this section, we conduct several experiments on both synthetic and real data to demonstrate the effectiveness and efficiency of OPIT and its variant OPITd. Their performance is evaluated in comparison with state-of-the-art algorithms. Our simulations are implemented using MAT-LAB on a laptop of Intel core i7 and 16GB of RAM. Our codes are also available online at https://github.com/thanhtbt/sst/ to facilitate replicability and reproducibility.

Experiments with Synthetic Data

Experiment Setup

Following the formulation in section 4.2, data samples {x t } t≥1 are generated at random under the standard model:

x t = A t w t + σ n n t , (4.30)
where n t ∈ R n×1 is a noise vector derived from N (0, I n ), σ n > 0 is to control the effect of the noise on algorithm's performance, w t ∈ R r×1 is an i.i.d. Gaussian random vector of zero-mean and unit-variance to represent the subspace coefficient. The sparse mixing matrix A t ∈ R n×r at time t is simulated as

A t = Ω ⊛ (A t-1 + εN t ), (4.31)
where ⊛ denotes the Hadamard product, Ω ∈ R n×r is a Bernoulli random matrix with probability 1 -ω sparse , N t is a normalized Gaussian white noise matrix, and ε > 0 is the time-varying factor aimed to control the subspace variation with time.

In order to evaluate the subspace estimation performance, we measure the following distance between two subspaces 7

d t ∆ = sin θ(A t , U t ), (4.32)
where U t refers to the estimated subspace at time t. 7 Given two orthonormal matrices A and U of the same size, we always have sin θ

(A, U) = ∥A ⊤ ⊥ U∥2 = ∥U ⊤ ⊥ A∥2 = ∥AA ⊤ -UU ⊤ ∥2
where (.) ⊥ denotes the orthogonal complement, e.g., U ⊤ U ⊥ = 0. In MATLAB, this distance can be easily computed by using the command sin(subspace(A, U)).

Effect of the forgetting factor β

The choice of the forgetting factor β plays an essential role in the tracking ability of OPIT. We investigated its effect by varying its value from 0.1 to 1 and then evaluating the performance of OPIT. Here, the data dimension, the true rank, the number of data samples were set at n = 50, r = 10, and T = 1000, respectively. We fixed the noise factor at σ n = 10 -3 , while two time-varying levels were considered, namely ε = 0 (stationary) and ε = 10 -3 (nonstationary).

Results are illustrated in Fig. 4.1. In the stationary environment (Fig. 1(a)), we can see that the higher the value of β is, the better the performance OPIT achieves, and β = 1 offers the best tracking performance. In the time-varying environment (Fig. 1(b)), 0 ≪ β < 1 can provide reasonably high subspace estimation accuracy. When β is close to 0, OPIT can track the underlying subspace over time but its accuracy is low. When β = 1, OPIT's performance degrades as time passes.

OPIT in Noisy and Dynamic Environments

In order to demonstrate the tracking ability of OPIT in nonstationary environments, we varied the value of the noise level σ n and the time-varying factor ε among {10 -1 , 10 -2 , 10 -3 } and then evaluated its subspace estimation accuracy. Two case studies were considered, including the small-scale {n = 100, r = 5} and the large-scale {n = 1000, r = 50} in which the sparsity level ω sparse was set to 90% and an abrupt change was created at t = 500. The forgetting factor β was fixed at 0.9 in both cases. We set the value of the thresholding factor k to ⌊10r log n⌉. on the performance of OPIT, respectively. We can see that the value of σ n and ε did not affect the convergence rate of OPIT but its estimation error. Despite the value of σ n and ε, OPIT still tracked successfully the underlying sparse subspace even in the presence of a significant change at t = 500. The lower σ n and ε are, the better subspace estimation accuracy OPIT can achieve.

Moreover, these experimental results indicate that the dimension n and rank r had in fact a small impact on how fast OPIT converges in dynamic environments. Specifically, when dealing with the large-scale setting, its convergence rate was faster than that when handling the small-scale one.

OPIT versus Other SST Methods

In this task, we compare the performance of OPIT against the state-of-the-art subspace tracking algorithms in different scenarios. These SST algorithms include ℓ 1 -PAST [START_REF] Yang | Fast STAP method based on PAST with sparse constraint for airborne phased array radar[END_REF], SS-FAPI [START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF],

SSPCA [START_REF] Yang | Streaming Sparse Principal Component Analysis[END_REF], and AdaOja [START_REF] Allen-Zhu | First efficient convergence for streaming k-PCA: A global, gap-free, and near-optimal rate[END_REF].

We used 1000 snapshots derived from the model (4.30) in which the time-varying factor ε was fixed at 10 -3 and the value of σ n was set to two levels: 10 -1 and 10 -3 . Here, two sparsity levels were also investigated, including 50% and 90%. The length of window was set to W = ⌊log n⌉ for the large-scale settings and low noise levels, while we used W = 1 for others. We fixed the forgetting factor β at 0.97 for all simulations in this task. For OPIT, the normalization step was used instead of the QR factorization. Parameters of other SST algorithms were kept default to have a fair comparison. When dealing with high-dimensional and large-scale settings, OPIT completely outperformed other SST algorithms at both low and high levels of noise as well as sparsity, as shown in Fig. 4.5.

SSPCA failed to track the underlying subspace while AdaOja, ℓ 1 -PAST, and SS-FAP could work in high dimensions. However, their performance in terms of estimation accuracy and convergence rate were much less than that of OPIT.

OPITd versus OPIT

We here investigate the tracking ability of OPITd in comparison with the original OPIT with respect to aspects: runtime, estimation accuracy, and robustness to abrupt changes.

To measure how fast OPITd is, we tested many configurations of {n, r} and reported its run time. Most other parameters were kept fixed as in the previous task except the number of snapshots T , including the sparsity level ω sparse = 90%, the noise level σ n = 10 -3 , the time- projection deflation might lead to a cumulative error between successive estimates. However, if the value of r is not too large, OPITd could track successfully the underlying subspace over time when the sparsity level ω sparse was not too high, as shown in Fig. 4.8.

Experiments with Real Video Data

In this task, four different video sequences are used to illustrate the effectiveness and efficiency of OPIT for real data, including "Lobby", "Hall", "Highway", and "Park" whose details are reported in Tab. 1, (see Fig. 4.9 for an illustration). We here compared the video tracking ability of OPIT with the state-of-the-art subspace tracking algorithms (i.e., ℓ 1 -PAST, SS-FAPI, and PETRELS-4.5. EXPERIMENTS ADMM [25]) and tensor tracking algorithms (i.e., SOAP [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF], OLCP [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF], OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], and ROLCP [33]). In order to apply these subspace tracking algorithms to the video sequences, each video frame of size I × J was reshaped into a IJ × 1 vector. Following the studies on video tracking in [25] and [33], the tensor rank and subspace rank were set to 10 for all simulations.

[ 1 0 , 2 ] [ 1 0 , 5 ] [ 2 0 , 5 ] [ 2 0 , 1 0 ] [ 5 0 , 5 ] [ 5 0 , 1 0 ] [ 1 0 0 , 1 0 ] [ 1 0 0 , 2 0 ] [ 5 0 0 , 1 0 ] [ 5 0 0 , 1 0 ] [ 1 0 0 0 , 5 0 ] [ 1 0 0 0 , 1 0 0 ] 0 100 200 300 OPIT OPITd [ 1 0 , 2 ] [ 1 0 , 5 ] [ 2 0 , 5 ] [ 2 0 , 1 0 ] [ 5 0 , 5 ] [ 5 0 , 1 0 ] [ 1 0 0 , 1 0 ] [ 1 0 0 , 2 0 ]
Simulation results are shown statistically in Tab. 7.1 and graphically in Fig. 4.10. As can be seen that OPIT provided a competitive estimation accuracy as compared to PETRELS-ADMM while its runtime was much faster than that of the ADMM-based tracking algorithm. Indeed, OPIT had a better performance than PETRELS-ADMM on the "Lobby" data, see Fig. Also, OPIT outperformed most tracking algorithms, apart from PETRELS-ADMM. With respect to runtime, ROLCP was the fastest "one-pass" tracking algorithm, several times faster than the second-best. Interestingly, our algorithm is also designed for handling a block of multiple incoming samples at each time (i.e. the length of window W > 1). When W = ⌊log(IJ)⌉, OPIT was even faster than ROLCP while still retaining a reasonable video tracking accuracy.

Conclusions

In this chapter, we have proposed a new provable OPIT algorithm which is fully capable of tracking the sparse principal subspace over time in both classical regime and high-dimension,

Appendix

Appendix A: Proof of Lemma 1

Because U t,F is the Q-factor of S t , we obtain θ(A, U t,F ) = θ(A, S t ) and hence

tan θ(A, U t,F ) = max ∥v∥ 2 =1 f (v) = A ⊤ ⊥ S t v 2 A ⊤ S t v 2 . (4.33)
For any vector v ∈ R r×1 and ∥v∥ 2 = 1, we can rewrite f (v) in (4.33) as follows

f (v) = A ⊤ ⊥ C + ∆C t U t-1 v 2 A ⊤ C + ∆C t U t-1 v 2 = A ⊤ ⊥ σ 2 x AA ⊤ + σ 2 n I N + ∆C t U t-1 v 2 A ⊤ σ 2 x AA ⊤ + σ 2 n I N + ∆C t U t-1 v 2 (i) = σ 2 n A ⊤ ⊥ U t-1 v + A ⊤ ⊥ ∆C t U t-1 v 2 σ 2 x + σ 2 n A ⊤ U t-1 v + A ⊤ ∆C t U t-1 v 2 (ii) ≤ σ 2 n A ⊤ ⊥ U t-1 2 + A ⊤ ⊥ ∆C t U t-1 2 (σ 2 x + σ 2 n ) A ⊤ U t-1 2 -A ⊤ ∆C t U t-1 2 (iii) ≤ σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 (σ 2 x + σ 2 n ) 1 -∥A ⊤ ⊥ U t-1 ∥ 2 2 -∥∆C t ∥ 2 . (4.34)
Here, (i) is due to A ⊤ ⊥ A = 0 (orthogonal complement); (ii) uses the inequality ∥P∥ 2 -∥Q∥ 2 ≤ ∥P + Q∥ 2 ≤ ∥P∥ 2 + ∥Q∥ 2 , ∀P, Q of the same size; and (iii) is derived from the following facts:

4.7. APPENDIX ∥P∆C t ∥ 2 ≤ ∥P∥ 2 ∥∆C t ∥ 2 , ∥A∥ 2 = ∥A ⊥ ∥ 2 = ∥U t-1 ∥ 2 = 1, and λ 2 min (A ⊤ U t-1 ) + λ 2 max (A ⊤ ⊥ U t-1 ) = 1, (4.35)
where λ max (P) and λ min (P) represent the largest and smallest singular value of P, respectively.

Indeed, the relation (4.35) leads to

A ⊤ U t-1 2 = λ max (A ⊤ U t-1 ) ≥ λ min (A ⊤ U t-1 ) = 1 -λ 2 max A ⊤ ⊥ U t-1 = 1 -A ⊤ ⊥ U t-1 2 2 , (4.36)
and thus, (iii) follows.

In parallel, it is well known that

sin ψ = 1 1 + tan -2 ψ ∀ψ ∈ [0, π/2] and h(x) = 1/ √ 1 + x -2
is an increasing function in the domain (0, ∞), i.e. x 1 ≤ x 2 implies h(x 1 ) ≤ h(x 2 ). Accordingly, we obtain

A ⊤ ⊥ U t,F 2 ≤ 1 1 + max v f (v) -2 = σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 σ 2 x + σ 2 n 1 -A ⊤ ⊥ U t-1 2 -∥∆C t ∥ 2 2 + + σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 2 1/2 . ( 4.37)
It ends the proof.

Appendix B: Proof of Lemma 2

We first recast U ⊤ t,⊥ U t,F 2 into the following form

U ⊤ t,⊥ U t,F 2 = U ⊤ t,F ,⊥ U t 2 = U ⊤ t,F ,⊥ U t -U t,F 2 = U ⊤ t,F ,⊥ ∆U t 2 . (4.38)
Under the following condition

(1 + √ 2)κ(S t ) S t -Ŝt F < S t
U ⊤ t,F ,⊥ ∆U t 2 ≤ U ⊤ t,F ,⊥ ∆U t F (i) ≤ κ(S t ) ∥U ⊤ t,F ,⊥ ∆S t ∥ F ∥S t ∥ 2 1 -(1 + √ 2)κ(S t ) ∥∆S t ∥ F ∥S t ∥ 2 (ii) ≤ ∥∆S t ∥ F λ min (S t ) -(1 + √ 2)∥∆S t ∥ F . (4.40)
Here, (i) follows immediately the perturbation theory for QR decomposition [START_REF] Chang | On the Perturbation of the Q-factor of the QR Factorization[END_REF]Theorem 3.1] and (ii) is obtained from the facts that ∥U t,F ,⊥ ∥ 2 = 1, ∥PQ∥ F ≤ ∥P∥ 2 ∥Q∥ F , and

∥P # ∥ 2 = λ -1 min (P) ∀P, Q of suitable sizes.
We also know that there always exists two coefficient matrices

H t ∈ R r×r and K t ∈ R (n-r)×r satisfying U t-1 = AH t + A ⊥ K t (i.
e. projection of U t-1 onto the subspace A) and

λ max (H t ) = ∥A ⊤ U t-1 ∥ 2 , λ min (H t ) = 1 -∥A ⊤ ⊥ U t-1 ∥ 2 2 , (4.41)
λ max (K t ) = ∥A ⊤ ⊥ U t-1 ∥ 2 , λ min (K t ) = 1 -∥A ⊤ U t-1 ∥ 2 2 .
Accordingly, we can express S t by

S t = CU t-1 + ∆C t U t-1 = (AΣ x A ⊤ + σ 2 n I n )(AH t + A ⊥ K t ) + ∆C t U t-1 = A(σ 2 x I r + σ 2 n I r )H t + σ 2 n A ⊥ K t + ∆C t U t-1 . (4.42)
Thanks to the fact that λ i (P + Q) ≥ λ i (P) -λ max (Q) ∀P, Q of the same size, the lower bound on λ min (S t ) is given by

λ min (S t ) ≥ λ min (σ 2 x + σ 2 n )AH t -λ max σ 2 n A ⊥ K t -λ max ∆C t U t-1 ≥ (σ 2 x + σ 2 n )λ min (H t ) -σ 2 n λ max (K t ) -∥∆C t ∥ 2 = (σ 2 x + σ 2 n ) 1 -∥A ⊤ ⊥ U t-1 ∥ 2 2 -σ 2 n ∥A ⊤ ⊥ U t-1 ∥ 2 -∥∆C t ∥ 2 , (4.43)
In what follows, we derive an upper bound on ∥∆S t ∥ F . For short, let us denote the support of A, U t-1 , and U t by T A , T t-1 , and T t , respectively, and

S t = T A ∪ T t-1 ∪ T t .
∆S t F = S t,St -S t,Tt F (i) ≤ ∥S t,St -S t,T A F = σ 2 n A ⊥ K t + ∆C t U t-1 F ≤ √ r σ 2 n A ⊥ K t + ∆C t U t-1 2 ≤ √ r σ 2 n ∥K t ∥ 2 + ∥∆C t ∥ 2 = √ r σ 2 n ∥A ⊤ ⊥ U t-1 ∥ 2 + ∥∆C t ∥ 2 , (4.44)
where (i) is due to

|T t | ≥ |T A | ∀t (i.e. |S t \ T t | ≤ |S t \ T A |), thanks the thresholding operator τ (.) with k/n ≥ ω sparse .
In parallel, we can rewrite the sufficient and necessary condition (4.39) as

(1 + √ 2) S # t 2 ∆S t F ≤ 1. (4.45) Since ∥S # t ∥ 2 = λ -1 min (S t ),
σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 (σ 2 x + σ 2 n ) 1 -A ⊤ ⊥ U t-1 2 ≤ √ 2 -1 √ r -1 + √ 2 . (4.46)
Under the condition (4.46), the upper bound on

∥U ⊤ t,⊥ U t,F ∥ 2 is U ⊤ t,⊥ U t,F 2 ≤ √ r σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 σ 2 x + σ 2 n 1 -A ⊤ ⊥ U t-1 2 2 -σ 2 n A ⊤ ⊥ U t-1 2 - -∥∆C t ∥ 2 - √ r 1 + √ 2 σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 = √ r σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 σ 2 x + σ 2 n 1 -A ⊤ ⊥ U t-1 2 2 -1 + √ r(1 + √ 2) × × σ 2 n A ⊤ ⊥ U t-1 2 + ∥∆C t ∥ 2 , (4.47)
thanks to (4.40). It ends the proof.

Appendix C: Proof of Lemma 3

We begin the proof with the following proposition:

Proposition 10. Given two sets of random variable vectors {a i } N i=1 and {b i } N i=1 where

a i i.i.d. ∼ N (0, σ 2 a I n ), b i i.i.d.
∼ N (0, σ 2 b I m ), and a i is independent of b j , ∀i, j. The following inequality holds 4.7. APPENDIX with a probability at least 1 -δ:

1 N N i=1 a i b ⊤ i 2 ≤ Cσ a σ b log(2/δ) max{n, m} N . (4.48)
where 0 < δ ≪ 1 and C > 0 is a universal positive number.

Proof. Its proof follows immediately Lemma 15 in [START_REF] Mitliagkas | Memory limited, streaming PCA[END_REF].

Since x i = Aw i + n i , we always have

∆C t 2 = 1 tW tW i=1 x i x ⊤ i -C 2 = 1 tW tW i=1 Aw i w ⊤ i A ⊤ + n i n ⊤ i + Aw i n ⊤ i + n i w ⊤ i A ⊤ -σ 2 x AA ⊤ -σ 2 n I n 2 ≤ A 1 tW tW i=1 w i w ⊤ i -σ 2 x I r A ⊤ 2 + 1 tW tW i=1 n i n ⊤ i -σ 2 n I N 2 + 2 A 1 tW tW i=1 w i n ⊤ i 2 ≤ A 2 2 1 tW tW i=1 w i w ⊤ i -σ 2 x I r 2 + 1 tW tW i=1 n i n ⊤ i -σ 2 n I n 2 + 2 A 2 1 tW tW i=1 w i n ⊤ i 2 , (4.49)
thanks to the inequality ∥PQ∥ 2 ≤ ∥P∥ 2 ∥Q∥ 2 for all P and Q of suitable sizes. Accordingly, with a probability at least 1 -δ (0 < δ ≪ 1), three components in the right hand side of (4.49)

are respectively bounded by

1 tW tW i=1 w i w ⊤ i -σ 2 x I r 2 ≤ C 1 log(2/δ)σ 2 x r tW , (4.50
)

1 tW tW i=1 n i n ⊤ i -σ 2 n I n 2 ≤ C 2 log(2/δ)σ 2 n n tW , (4.51
)

1 tW tW i=1 w i n ⊤ i 2 ≤ C 3 log(2/δ)σ x σ n n tW , (4.52)
where C 1 , C Thanks to Lemma 3, when t satisfies (4.18), i.e.,

t ≥ C log(2/δ)r 2 W ϵ 2 ρ 2 √ r + σ 2 n σ 2 x + 2 σ n σ x √ n 2 , (4.54)
we obtain

∥∆C t ∥ 2 ≤ r -1 ρσ 2 x ϵ with 0 < ρ ≤ r.
In what follows, two case studies d t-1 ≥ ϵ and

d t-1 ≤ ϵ are investigated. Case 1: When d t-1 ≥ ϵ, i.e., ∥∆C t ∥ 2 ≤ r -1 ρσ 2 x d t-1 .
We can rewrite ∥A ⊤ ⊥ U t,F ∥ 2 as follows

A ⊤ ⊥ U t,F 2 ≤ (σ 2 n + r -1 ρσ 2 x )d t-1 (σ 2 n + σ 2 x ) 1 -d 2 t-1 -r -1 ρσ 2 x d t-1 2 + aaaaaaaaaaaa + (σ 2 n + σ 2 x ρ/r) 2 d 2 t-1 1/2 (i) ≤ (σ 2 n + r -1 ρσ 2 x )d t-1 (σ 2 n + σ 2 x ) 1 -ω 2 0 -r -1 ρσ 2 x ω 0 2 + aaaaaaaaaaaaaaaaaaaaa + (σ 2 n + r -1 ρσ 2 x ) 2 ω 2 0 1/2 (ii) ≤ (σ 2 n + r -1 ρσ 2 x )d t-1 (1 + γ 2 r 2 )σ 4 n + (1 -ργ) 2 σ 4 x + aaaaaaaa + 2(1 -ργ + γ 2 r 2 )σ 2 x σ 2 n 1/2 1 -ω 2 0 . (4.55)
Here, (i) is obtained from the fact that g(x) = (a

√ 1 -x 2 -bx) 2 + cx 2 -1/2 is an increasing function in the range [0, √ 2/2]
where a, b, and c are defined therein 8 and (ii) is simple due to the fact that there always exists a small parameter γ > 0 such that ργ < 1 and ω 0 ≤ γr 1 -ω 2 0 .

In the similar way, we obtain the following upper bound on ∥U ⊤ t,⊥ U t,F ∥ 2 :

8 Writing x = sin y, the domain of y is [0, π/4].

Here, we can recast g(x) into g(y) = (a cos y -b sin y) 2 + c sin 2 y -1/2 . The derivative g ′ (y) is given by

g ′ (y) = 0.5 (a cos y -b sin y) 2 + c sin 2 y -3/2 (a 2 -b 2 -c) sin(2y) + ab cos(2y) . Since a 2 -b 2 > c by their definition, g ′ (y) > 0 ∀y ∈ [0, π/4] and hence g ′ (x) = g ′ (y)dy/dx = g ′ (y)/ √ 1 -x 2 > 0 ∀x ∈ [0, √ 2/2]. Accordingly, dt-1 ≤ ω0 ≤ √ 2/2 implies g(dt-1) ≤ g(ω0) which (i) then follows. 4.7. APPENDIX U ⊤ t,⊥ U t,F 2 ≤ √ r σ 2 n + r -1 ρσ 2 x d t-1 σ 2 x + σ 2 n 1 -d 2 t -1 + √ r(1 + √ 2) × × σ 2 n + r -1 ρσ 2 x d t-1 (i) ≤ √ r(σ 2 n + r -1 ρσ 2 x )d t-1 σ 2 x + σ 2 n 1 -ω 2 0 -1 + √ r(1 + √ 2) σ 2 n + r -1 ρσ 2 x ω 0 (ii) ≤ √ r(σ 2 n + r -1 ρσ 2 x ) (σ 2 x + σ 2 n )(1 -ϱ) 1 -ω 2 0 d t-1 , (4.56)
where

ϱ = γ 1 + √ r(1 + √ 2)(rσ 2 n + ρσ 2 x ) (σ 2 x + σ 2 n ) -1 . Specifically, (i) is due to the increasing property of z(x) = (a √ 1 -x 2 -bx) -1
, and (ii) thanks to ω 0 ≤ γr 1 -ω 2 0 .

Thanks to (4.55) and (4.56), we obtain

d t ≤ A ⊤ ⊥ U t,F 2 + U ⊤ t,⊥ U t,F 2 ≤ rσ 2 n + ρσ 2 x rξ 1 -ω 2 0 d t-1 , (4.57)
where

ξ = 0.5 max (1 + γ 2 r 2 )σ 4 n + (1 -ργ) 2 σ 4 x + 2(1 -ργ + γ 2 r 2 )σ 2 x σ 2 n 1/2 , (σ 2 x + σ 2 n )(1 -ϱ)/ √ r . (4.58)
Note that in order to utilize the two bounds (6.189) and (4.56), the condition (4.46) must be satisfied which is equivalent to

(σ 2 n + r -1 ρσ 2 x )ω 0 (σ 2 x + σ 2 n ) 1 -ω 2 0 ≤ √ 2 -1 √ r -1 + √ 2 . (4.59)
Accordingly, we obtain ω 0 ≤ α(r,ρ)

1-α(r,ρ) 1/2 where α(r, ρ) = (3 -2 √ 2)(σ 2 x + σ 2 n ) 2 r + 2 √ r( √ 2 -1) + 3 -2 √ 2 σ 2 n + r -1 ρσ 2 x 2 . (4.60) In parallel, α(r, ρ) ≥ 3-2 √ 2 r+2 √ r( √ 2-1)+3-2 √
2 for every 0 < ρ ≤ r. Thus, we obtain

ω 0 ≤ 3 -2 √ 2 r + 2 √ r( √ 2 -1)
t ≥ log(ϵ/ω 0 ) W log(rσ 2 n + ρσ 2 x ) -log(rξ 1 -ω 2 0 ) . (4.63)
which is lower than the bound (4.18). Therefore, we can conclude that d t ≤ ϵ holds and it ends the proof. Tensor decomposition has been demonstrated to be successful in a wide range of applications, from neuroscience and wireless communications to social networks. In an online setting, factorizing tensors derived from multidimensional data streams is however non-trivial due to several inherent problems of real-time stream processing. In recent years, many research efforts have been dedicated to developing online techniques for decomposing such tensors, resulting in significant advances in streaming tensor decomposition or tensor tracking. This topic is emerging and enriches the literature on tensor decomposition, particularly from the data stream analystics perspective. Thus, it is imperative to carry out an overview of tensor tracking to help researchers and practitioners understand its development and achievements, summarise the current trends and advances, and identify challenging problems. In this article, we provide a contemporary and comprehensive survey on different types of tensor tracking techniques. We particularly categorize the state-of-the-art methods into three main groups: streaming CP decompositions, streaming Tucker decompositions, and streaming decompositions under other tensor formats (i.e., tensor-train, t-

SVD, and BTD). In each group, we further divide the existing algorithms into sub-categories based on their main optimization framework and model architectures. Finally, we present several research chal-

lenges, open problems, and potential directions of tensor tracking in the future.

Introduction

Tensor decomposition (TD) has attracted much attention from the signal processing and machine learning community [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF]. As a tensor is a multiway array, it provides a natural representation for multidimensional data. Accordingly, TD which factorizes a tensor into a set of basis components (e.g., vectors, matrices, or simpler tensors) has become a popular tool for multivariate and high-dimensional data analysis. In particular, we have witnessed significant advances in TD and a rapid growth in its applications over the last two decades [START_REF] Liu | Tensor Computation for Data Analysis[END_REF]. Several types of TD, such as CANDECOMP/PARAFAC (CP) [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF], high-order SVD (HOSVD)/Tucker [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF], tensor train/network [START_REF] Oseledets | Tensor-train decomposition[END_REF], t-SVD [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF], and block-term decomposition (BTD) [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF], have been developed and successfully applied to various domains, from neuroscience [START_REF] Thanh | Multi-channel EEG epileptic spike detection by a new method of tensor decomposition[END_REF][START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF] wireless communications [START_REF] Nion | Tensor algebra and multidimensional harmonic retrieval in signal processing for MIMO radar[END_REF][START_REF] Chen | Tensor decompositions in wireless communications and MIMO radar[END_REF] to social networks [START_REF] Nakatsuji | Semantic social network analysis by cross-domain tensor factorization[END_REF][START_REF] Fernandes | Tensor decomposition for analysing timeevolving social networks: An overview[END_REF].

INTRODUCTION

The demand for (near) real-time stream processing has been increasing over the years since many modern applications (e.g., Internet-of-Things) generate massive amounts of streaming data over time and analytical insights from such data can bring several benefits, e.g., for real-time decision making [2]. As its name implies, (near) real-time stream processing needs to immediately deliver and analyse data streams upon their arrival. Since streaming data grow bigger, faster, and become more complex by the time, there exist several inherent problems which are still challenging issues, such as (i) the unbounded size of streaming data, (ii) time-varying model, concept drift, or dataset shift, and (iii) uncertainty and imperfection, etc. We refer the readers to [2,[START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF] for good surveys on data stream analysis.

When using tensors to represent data streams, TD is generally referred to as tensor tracking or adaptive/online/ streaming tensor decomposition. Specifically, factorizing a streaming tensor is nontrivial due to several computational challenges. First, as tensor streams are continuously generated, their volume grows significantly over time and possibly to infinity. Applying the conventional batch TD methods to such tensors is not possible as they require data to be stored and processed offline. Second, properties of streaming tensors (e.g., the low-rank approximation model) can vary with time in unforeseen ways. Moreover, tensor streams often happen in real-time, so retransmission of a stream is difficult, even impossible. Accordingly, batch tensor estimation and decomposition become less accurate when time passes. Last but not least, some modern applications require high-speed data acquisition systems to rapidly acquire and process massive data streams. In such a case, very fast and (near) real-time processing is highly important. However, batch TDs are often of high complexity, and hence turn out to be inefficient. These characteristics make tensor tracking much different from batch tensor decomposition and lead to several distinguishing requirements for tensor trackers, such as low latency and memory storage, high scalability, adaptation to time variation, and robustness, to name a few.

As the literature of tensor tracking has significantly expanded in recent years, it is imperative to it is imperative to conduct an extensive overview of the state-of-the-art tensor tracking algorithms to help researchers and practitioners to identify: (i) which topics in tensor tracking are significant and emerging, (ii) what kind of tracking models and related analysis techniques have already been deployed to date and how to apply them in specific tasks, and (iii) main research challenges, open problems, and potential directions of tensor tracking in the future.

State-of-the-art Surveys

The very first and gentle introduction to tensor and tensor decomposition was provided by Rasmus in [START_REF] Bro | PARAFAC. Tutorial and applications[END_REF] two decades ago. This reference offered a tutorial on CP/PAFRAFAC decomposition covering features, properties, methods, and applications in chemometrics. Since then, there have been many published survey papers which provided different points of view on tensor computation in the literature. We can broadly divide them into three classes, including (i) surveys A literature survey on unsupervised multiway data analysis: multiway models (i.e., CP family and Tucker family), their workhorse algorithms and applications.

[10] (2009)

CP & Tucker decomposition

An extensive survey on main algorithms, properties and applications of CP, Tucker decompositions and their variants.

A list of software and toolboxes for tensor processing.

[187] (2010) Tucker/HOSVD decomposition An overview on numerical methods for Tucker/HOSVD decomposition & its applications in signal processing.

[188] (2013) Low-rank tensor approximations A literature survey on low-rank tensor approximation models and algorithms.

[189] (2014) Incomplete tensor decomposition A survey on numerical methods for factorizing incomplete tensors and their connections to signal processing applications.

[12] (2016) Tensor network decomposition An extensive tutorial on tensor networks, their operations and algorithms.

[190] (2016) Big tensor decomposition A brief review of methods for factorizing large-scale tensors.

[191] (2020) Tucker/HOSVD decomposition A survey on randomized algorithms for computing Tucker/HOSVD decomposition. [START_REF] Fu | Computing large-scale matrix and tensor decomposition with structured factors: A unified nonconvex optimization perspective[END_REF] A list of software and toolboxes for tensor processing. [START_REF] Zare | Extension of PCA to higher order data structures: An introduction to tensors, tensor decompositions, and tensor PCA[END_REF] (2018) Connections to PCA An introduction to tensors and tensor decompositions in the lens of PCA.

Surveys on tensor applications [START_REF] Mørup | Applications of tensor (multiway array) factorizations and decompositions in data mining[END_REF] (2011) Data analysis An overview of tensor decomposition applications for a wide variety of data and problem domains. [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF] (2015)

Signal processing A comprehensive survey on tensor decompositions for signal processing. [START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF] (2015)

EEG applications A brief survey on tensor decompositions of EEG signals. [START_REF] Fanaee-T | Tensor-based anomaly detection: An interdisciplinary survey[END_REF] (2016)

Anomaly detection An interdisciplinary survey on tensor-based anomaly detection. [START_REF] Papalexakis | Tensors for data mining and data fusion: Models, applications, and scalable algorithms[END_REF] (2017) Data fusion A review of tensor decompositions with emphasis on data fusion applications.

[202] (2017)

Machine learning & data analysis

An tutorial on tensor network models for super-compressed representation of data and their applications in machine learning and data analytics. [START_REF] Ji | A survey on tensor techniques and applications in machine learning[END_REF] (2019)

Machine learning An overview of tensor techniques and applications in machine learning.

[204] (2021) Multisensor signal processing A comprehensive survey on tensor methods for multisensor signal processing.

[182] (2021) Wireless communications A comprehensive overview of tensor decompositions for wireless communications.

[184] (2021) Social networks A survey on tensor decomposition for analysing time-evolving social networks.

[205] (2021)

Computer vision & deep learning

A practical overview of tensor methods for computer vision and deep learning [START_REF] Batselier | Low-rank tensor decompositions for nonlinear system identification: A tutorial with examples[END_REF] (2022) Nonlinear system identification A tutorial on tensor methods for nonlinear system identification.

[13] (2022) Data analysis A systematic and up-to-date overview of tensor decompositions from the engineer's point of view.

This work

Streaming tensor decomposition (Tensor tracking)

A contemporary and comprehensive survey on methods for factorizing tensors derived from data streams under several tensors formats.

Research challenges, open problems, and future directions.

on models, methods, and tools for factorizing tensors, (ii) surveys on general tensor problems, e.g., tensor operations, uniqueness, ranks, filtering, spectral analysis, and complexity, and (iii) surveys on tensor applications. We refer the readers to Tab. 5.1 for the main contributions of the state-of-the-art surveys on tensors.

Among them, the most notable and highly-cited survey paper is the work of Kolda et al.

in [START_REF] Kolda | Tensor decompositions and applications[END_REF] that was published in the SIAM Review journal more than a decade ago. The survey 5.1. INTRODUCTION presented basic multiway models (i.e., CP family and Tucker family) and workhorse algorithms for factorizing tensors under these models. Some applications and software for tensors were also mentioned. The second key survey in the literature is the work of Sidiropoulos et al. in [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF] that appeared five years ago in the IEEE Transactions on Signal Processing journal. To fill some gaps in the existing surveys on CP and Tucker decompositions of that time, the authors provided an in-depth overview of tensors with respect to the following aspects: uniqueness, ranks, bounds, algorithms, and applications. Moreover, an up-to-date list of software and toolboxes for tensor computation was provided therein. To extend beyond the two standard multiway models, Cichocki et al. conducted a comprehensive tutorial on tensor networks in [START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions[END_REF][START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 2 applications and future perspectives[END_REF] that appeared in the Foundations and Trends in Machine Learning journal. Particularly, its coverage includes tensor network models, the associated operations and algorithms, and their applications. Besides, it also highlighted connections of tensor networks to dimensionality reduction and large-scale optimization problems. Very recently, Liu et al. provided a general overview of tensors from the engineer's point of view in the book Tensor Computation for Data Analysis [START_REF] Liu | Tensor Computation for Data Analysis[END_REF]. It covers various aspects of tensor computations and decompositions, from operations and well-known multiway representations to tensor-based data analysis techniques and practical applications.

However, to date, we are not aware of any survey paper specifically reviewing the problem of streaming tensor decomposition or tensor tracking. Therefore, it is of great interest to carry out an overview of this topic to enrich the tensor literature.

Main Contributions

In this chapter, we present a contemporary and comprehensive survey on the state-of-the-art online techniques which are capable of factorizing tensors derived from data streams.

Our survey begins with basic coverage of five common tensor decompositions and their main features. They are CP/PARAFAC, HOSVD/Tucker, BTD, tensor-train, and t-SVD. Two kinds of streaming models are then introduced to represent streaming tensors, including single-aspect and multi-aspect. Next, we review four main groups of streaming CP decomposition algorithms:

Tensor Decompositions

In this section, we briefly describe the background of the five popular tensor decompositions which have already been deployed to factorize streaming tensors in an online fashion. They are CP/PARAFAC, HOSVD/Tucker, BTD, tensor-train, and t-SVD.

TENSOR DECOMPOSITIONS

CP/PARAFAC Decomposition

Under the CP format [START_REF] Bro | PARAFAC. Tutorial and applications[END_REF], a tensor

X ∈ R I 1 ×I 2 ו••×I N can be decomposed into a set of N matrices {U (n) } N n=1
sharing the same number of columns as follows

X ∆ = {U (n) } N n=1 = r i=1 U (1) (:, i) • U (2) (:, i) • • • • • U (N ) (:, i), (5.1)
where the so-called tensor factor U (n) is of size I n × r with 1 ≤ n ≤ N . The smallest r satisfying (5.1) is referred to as the CP-rank of X .

This decomposition has its advantages and disadvantages. On the one hand, CP is the best memory-saving format for representing high-order tensors, and hence, it can overcome the curse of dimensionality which particularly limits the order of tensors to be analysed. Under certain conditions, CP decomposition is essentially unique up to a permutation and scale which is an useful property in many applications, e.g., to recover exact components or individuals hidden in the underlying data. However, its main disadvantage is that finding the true CP-rank r is known as an NP-hard problem [START_REF] Hillar | Most tensor problems are NP-hard[END_REF]. Even though the CP-rank is given in advance, the best rank-r approximation of a tensor may not exist [START_REF] Silva | Tensor rank and the ill-posedness of the best low-rank approximation problem[END_REF]. To compute the CP decomposition, one of the most widely-used approaches is based on the alternating least-squares (ALS) technique [START_REF] Kolda | Tensor decompositions and applications[END_REF].

Tucker Decomposition

Under the Tucker format [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF], we can factorize X into a core tensor G of a smaller size and

N factors {U (n) } N n=1 as X ∆ = G; {U (n) } N n=1 = G × 1 U (1) × 2 U (2) × 3 • • • × N U (N ) , (5.2)
where G is of size r 1 × r 2 × • • • × r N with r n ≤ I n , and U (n) ∈ R In×rn is an orthogonal matrix.

The vector r = [r 1 , r 2 , . . . , r N ] is called the multilinear rank or rank-(r 1 , r 2 , . . . , r N ) of X .

Tucker decomposition is more flexible than CP in the sense that we can write any tensor X in the form (5.2) and its computation can be done effectively and stably. The two most popular algorithms for computing (5.2) are HOSVD and Higher-order Orthogonal Iteration (HOOI) [START_REF] Lathauwer | On the best rank-1 and rank-(r 1 , r 2 " r n ) approximation of higher-order tensors[END_REF].

Both HOSVD and HOOI offer a good rank-(r 1 , r 2 , . . . , r N ) tensor approximation for X and they can be efficiently implemented in practice. In general, the Tucker representation is not unique but the subspace covering U (n) is physically unique. Therefore, the main interest in Tucker decomposition is for finding subspaces of the tensor factors, and hence, for approximation, dimensionality reduction, and feature extraction [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF].

123 5.2. TENSOR DECOMPOSITIONS

Block-Term Decomposition

Block-term decomposition (BTD) allows to represent X as a sum of low multilinear-rank tensors [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF]:

X = R i=1 G i ; {U (n) i } N n=1 , (5.3)
where

{G r } R r=1 with G i ∈ R r 1 ×r 2 ו••×r N is the set of core tensors, U (n) = U (n) 1 , . . . , U (n) R with U (n) i
∈ R In×rn is the n-th tensor factor, and r n ≤ I n ∀ i, n.

The BTD format (5.3) can be considered as a combination of CP and Tucker. As its name reveals, the basic components in BTD are rank-(r 1 , r 2 , . . . , r N ) blocks while they are rank-1 terms in CP/PARAFAC and matrix decompositions. When these blocks are rank-1 tensors (i.e., r n = 1 ∀n), it boils down to CP. When it has only one block (i.e., R = 1), BTD becomes the standard Tucker decomposition. It is worth noting that the number of blocks R relies on the block's size. Like CP, BTD is essentially unique [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF]. The common approach to find (5.3) is also based on the ALS technique [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part III: Alternating least squares algorithms[END_REF].

Tensor-train Decomposition

Tensor-train (TT) decomposition expresses X as a multilinear product of third-order tensors

{G (n) } N n=1 according to X = G (1) × 1 1 G (2) × 1 2 • • • × 1 N G (N ) , (5.4)
where G (n) ∈ R r n-1 ×In×rn is the n-th TT-core (aka tensor carriage) with n = 1, 2, . . . , N . Here, r 0 = r N = 1 and the quantities {r n } N -1 n=1 are called TT-ranks [START_REF] Oseledets | Tensor-train decomposition[END_REF].

This type of TD offers several appealing benefits for representing tensors, especially high-order tensors. For instance, given an arbitrary tensor X , we always find a set of TT-cores {G (n) } N n=1 satisfying (5.4) with suitable TT ranks. Besides, its TT-ranks can be effectively estimated in a stable way in contrast to the CP-rank determination [START_REF] Hillar | Most tensor problems are NP-hard[END_REF]. Moreover, TT also offers a memorysaving representation for tensors and can break the curse of dimensionality like CP. With respect to the implementation, the workhorse algorithm to compute TT is TT-SVD [START_REF] Oseledets | Tensor-train decomposition[END_REF].

TENSOR TRACKING FORMULATION

T-SVD Decomposition

Tensor SVD (t-SVD) is another multiway extension of SVD for decomposing tensors in which X is factorized into three tensors U , G, and V of the same order:

X = U * G * V H , (5.5)
where " * " denotes the t-product, U and V are unitary tensors, and G is a rectangle f -diagonal tensor whose frontal slices are diagonal matrices [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF]. To define the low-rank tensor approximation under the t-SVD format, the so-called tubal-rank r t is determined as the number of non-zero tubes in G, (e.g., when the tensor X is of order 3,

r t (X ) = i 1[G(i, i, :) ̸ = 0] where 1 is an indicator function).
The t-SVD algebraic framework is quite different from the classical multilinear algebra in other types of TD. Thanks to the t-product and Fourier transform, several linear, multilinear operators and other transformations are successfully extended from matrices to tensors, such as transpose, orthogonality, and inverse. In particular, t-SVD can be effectively obtained by computing SVDs in Fourier domain and its performance (i.e., exact recovery with high probability) can be guaranteed under mild conditions [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF].

Tensor Tracking Formulation

In this section, the problem of tensor tracking is formulated. Specifically, we first divide streaming tensor models into two classes and then construct some terminologies to support the problem statement. They are single-aspect and multi-aspect streaming models, see Fig. 5.2 for an illustration. After that, we formulate a general formulation of the tensor tracking problem which is suitable for many applications.

Single-aspect Streaming Model

In the classical online setting, we are interested in the decomposition of an N -order streaming tensor X t fixing all but one dimension (mode). Without loss of generality, we suppose the last dimension is temporal, and hence, we can write

X t ∈ R I 1 ו••×I N -1 ×I t N where I t N is increasing with time.
The following definition of temporal slices is useful to formulate the problem of single-aspect tensor tracking.

Definition 1 (Temporal slice). Given a streaming tensor Without loss of generality, we assume that I t N = t meaning that at each time instant one new slice of the tensor is observed. Accordingly, the streaming tensor X t can be viewed as a set of temporal slices {Y τ } t τ =1 . In other word, X t is derived from appending the new comming temporal slice Y t to the previous observations X t-1 along the time dimension, i.e.,

X t ∈ R I 1 ו••×I N -1 ×I t N , we say Y τ = X t (:, . . . , :, τ ) ∈ R I 1 ×I 2 ו••×I N -1 is the τ -th temporal slice of X t for 1 ≤ τ ≤ I t N . 5.3. TENSOR TRACKING FORMULATION t 1  t 2  t Single-aspect Multi-aspect
X t = X t-1 ⊞ N Y t and I t N = I t-1 N + 1 = t. (5.6)
Generally, Y t has the form

Y t = P t ⊛ L t + N t + O t , (5.7)
where " ⊛ " denotes the Hadamard product, L t is a low-rank component, P t is a binary tensor, N t is a noise tensor, and O t is a sparse tensor. The data model (5.7) is a general form which is suitable for many scenarios. For example, P t represents missing and observed entries of Y t ;

N t is an additive white Gaussian noise; and O t denotes the sparse outliers. Meanwhile, the low-rank L t , which can be formulated by CP, Tucker, BTD, TT, or t-SVD format, can be static or time-varying. Based on these terminologies, the problem of single-aspect tensor tracking can be formally stated as follows:

Single-aspect Tensor Tracking: At time t, given a temporal slice Y t and old estimates of X t-1 (e.g., core tensors and tensor factors), we want to track the new estimates of

X t = X t-1 ⊞ N Y t in time.

TENSOR TRACKING FORMULATION

Multi-aspect Streaming Model

In some modern online applications, tensor data may evolve in multiple dimensions/modes over time, and thus, the single-aspect streaming model is not useful for modelling such streaming data. In [START_REF] Fanaee-T | Multi-aspect-streaming tensor analysis[END_REF], Fanaee-T et al. for the first time introduced the concept of multi-aspect streaming tensors to represent streaming data having more than one dimension increasing with time. Since then, some online algorithms have been developed to deal with the problem of multi-aspect streaming tensor decomposition.

For convenience, we first introduce the definitions of multi-aspect streaming tensors and temporal tubes.

Definition 2 (Multi-aspect streaming tensor). A set of N -order tensors {X t } t≥1 is called a multi-aspect streaming tensor sequence denoted as {X } when

X t ∈ R I t 1 ×I t 2 ו••×I t N , I t n = I t-1
n + W t n where W t n ≥ 0, 1 ≤ n ≤ N , and X t-1 is a sub-tensor of X t . If X t belongs to such a sequence {X }, we say that X t is a multi-aspect streaming tensor.

Definition 3 (Temporal tube). Given two successive tensors X t-1 and X t derived from the same multi-aspect streaming tensor sequence {X }, the coming data stream at time t can be represented by Y t = X t \X t-1 of the same size as X t with entries

Y t i 1 ,...,i N =    X t i 1 ,...,i N if I t-1 n < i n ≤ I t n , 0 otherwise, (5.8)
for 1 ≤ n ≤ N . We say that the non-zero entries in Y t are temporal tubes. Now, we can state the problem of multi-aspect tensor tracking as follows:

Multi-aspect Tensor Tracking: At time t, given temporal tubes in Y t , and old estimates of X t-1 (e.g., core tensors and tensor factors), we want to track the new estimates

of X t = X t-1 ∪ Y t in time.
It is worth noting that the single-aspect tensor tracking problem also belongs to the class of multi-aspect tensor tracking where most of the tensor dimensions I n are constant by setting W t n = 0, except the last one I t N . Besides, temporal slices may be regarded as frontal slices of the tensor Y t defined as in (5.8).

General Formulation of Optimization

We here provide a general formulation of tensor tracking which can be used in many applications.

In particular, the optimization problem can be written as

argmin {G},{U },O t τ =1 β τ ℓ Y τ , P τ , {G}, {U }, O Minimize residual errors + ρ G R G {G} Regularize cores + ρ U R U {U } Regularize factors + ρ O R O O Promote sparsity + λ G L G {G} + λ U L U {U } Orientate applications .
(5.9)

Here, {G} and {U } denote the set of core tensors and tensor factors respectively, while O is to represent data corruptions by impulsive noise or outliers. Specifically, the three terms in the second line of (6.5) are used to present regularizations or penalty terms imposed on parameters of interest. The last two penalty terms of (6.5) are for the application orientation. The main loss function ℓ(.) is defined to minimize the residual errors between the estimations and observations.

Streaming CP Decomposition

The primary objective of this section is to provide technical descriptions of the-state-of-the-art online techniques for factorizing streaming tensors under the CP format. In the literature, there are many streaming CP algorithms and they can be categorized into the following groups: (i)

subspace-based methods, (ii) block-coordinate descent methods, (iii) Bayesian inference, and (iv) multi-aspect streaming CP decompositions. The three former groups are particularly developed for single-aspect streaming models, while the latter is dedicated to the factorization of tensors having more than one temporally varying mode. The readers are referred to Tabs. 5.2 and 5.3

for quick comparisons of the existing streaming CP decomposition algorithms. In what follows, we take each group into consideration.

Subspace-based Methods

The very first study addressing the problem of streaming CP decomposition is of Nion and

Sidiropoulos in [START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF]. Specifically, the authors introduced the two novel adaptive CP algorithms called PARAFAC-SDT and PARAFAC-RLS capable of tracking third-order streaming tensors having one temporal dimension. Both algorithms are based on the subspace-based approach in which we first track a low-dimensional tensor subspace, and then recover the loading matrices from exploiting its Khatri-Rao structure. Following the same line, some other adaptive CP algorithms were proposed for tensor tracking such as CP-PETRELS [START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF], 3D-OPAST [START_REF] Nguyen | Fast adaptive PARAFAC decomposition algorithm with linear complexity[END_REF],

5.4. STREAMING CP DECOMPOSITION (1 ) t U 1 [ ] Ir  2 [] Ir  3 3 ( ) [ ] I d r  
New Observations and SOAP [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF]. In the following, we describe their subspace-based framework for factorizing streaming tensors with time.

I (3) t U 1 t  X (2) t U (1) t U  (1 ) t U 1 [] Ir  2 [] Ir  I (3) 1 t  U (2) 1 t  U (1) 1 t  U  3 [] Ir  (3) ˆt U 1 t  X 2 I 1 I 3 I 3 d  t X (
First, we recall that the low-rank L t of Y t has the form

L t = {U (n) t } N -1 n=1 , u (N ) t
, where u

(N ) t is the last row of U (N )
t . Thus, L t can be recast into the following form:

ℓ t ∆ = vec(L t ) = N -1 n=1 U (n) t u (N ) t ⊤ = H t u (N ) t ⊤ , (5.10)
where H t ∈ R I 1 ...I N -1 ×r plays a role as a mixing matrix while u (N ) t can be viewed as a coefficient vector in subspace tracking problems. Accordingly, streaming CP decomposition can boil down to a constrained problem of subspace tracking where the basis matrix has a Khatri-Rao structure.

Particularly for N = 3, the authors in [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF][START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF][START_REF] Nguyen | Fast adaptive PARAFAC decomposition algorithm with linear complexity[END_REF][START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF] proposed to solve the following objective function:

U (n) t 3 n=1 = argmin {U (n) } 3 n=1 t τ =1 β t-τ p τ ⊛ y τ -H u (3) τ ⊤ 2 2 s.t. H = U (1) ⊙ U (2) , (5.11)
where y τ = vec(Y τ ), p τ = vec(P τ ), and u τ is the τ -th row of the temporal factor U

t , and β is a forgetting factor aimed at discounting the impact of distant observations. Specifically, (5.11) can be effectively solved by applying the following procedure:

• Stage 1: Estimate H t and u (3) t , given old estimates of U (1) t-1 and U (2) t-1 ; • Stage 2: Find U (1) t , U (2)
t satisfying H t ≃ U (1) t ⊙U (2)
t , and then re-update

H t ← U (1) t ⊙U (2) t ; • Stage 3: Update U (3) t = U (3) t-1 ⊤ u (3) t ⊤ ⊤
where u

(3) t can be re-estimated as in Step 1 (optional).

STREAMING CP DECOMPOSITION

✓ ✗ ✗ ✗ random O r 2 |Ω| -BCD + SGD OLCP [175] ✗ ✗ ✓ ✗ ✓ O r 2 I N -1 -BCD + SGD SOAP [174] ✗ ✗ ✗ ✗ ✓ O rI 2 -Subspace-based + Second- order estimation -Supports nonnegativity CP-NLS [213] ✗ ✗ ✗ ✗ ✓ O(r 2 I 2 ) -Nonlinear least-squares BRST [214] ✓ ✓ ✓ ✗ ✓ unavailable -Variational Bayesian CP-PETRELS ✓ ✗ ✗ ✗ ✓ O r 2 |Ω| -Subspace-based [215] -Tracking using PETRELS CP-stream [216] ✗ ✗ ✓ ✗ random O r 2 I N -1 -ADMM + tuning-free -Supports sparsity POST [217] ✓ ✗ ✓ ✗ ✓ O r 3 N I N -1 -Variational Bayesian OLSTEC [176] ✓ ✗ ✗ ✓ random O r 2 I 2 -BCD + RLS iPARAFAC ✗ ✗ ✗ ✗ ✓ O r 2 |S| -Apache Spark a [218] |S| : size of the selected set -Randomized MTTKRP TensorNOODL ✗ ✗ ✗ ✓ ✓ O(r 2 I 2 )
-Online dictionary learning [START_REF] Rambhatla | Provable online CP /PARAFAC decomposition of a structured tensor via dictionary learning[END_REF] -Supports sparsity

SPADE [220] ✗ ✗ ✓ ✗ ✓ O(r 3 I N -1 ) -Streaming PARAFAC2 b SliceNStitch ✗ ✗ ✓ ✗ random O rN |S| + (rN ) 2 +N r 3 with |S| : number of non-zeros -Sparse decomposition [221] SOFIA [222] ✓ ✓ ✓ ✗ ✓ O r 3 I N -1 -Holt-Winters fitting c -Supports seasonality STF [223] ✓ ✗ ✓ ✗ ✓ O (N + r)N r|Ω| +N Ir 3 -BCD + SGD ACP [29, 33] ✓ ✗ ✓ ✓ random O r 2 |S| with |S| : size -Random sampling of the selected set -BCD + RLS RACP [27] ✓ ✓ ✓ ✓ random O r 2 I N -1 -ADMM + RLS -ℓ1-norm penalty Online CPDL ✗ ✗ ✓ ✓ ✓ O r 2 I N -1
-Nonnegative decomposition [START_REF] Lyu | Online nonnegative CP-dictionary learning for Markovian data[END_REF] -Markovian data -Online dictionary learning

⋆ Suppose that I1 = I2 = • • • = IN = I, rCP = r,
and |Ω| is the number of observed elements.

# Abbreviations: RLS (recursive least-squares), SDT (simultaneous diagonalization tracking), BCD (block-coordinate descent), ADMM (alternating direction method of multipliers), SGD (stochastic gradient descent), and MTTKRP (matricized-tensor times Khatri-Rao product).

a Apache Spark is a unified data analytics framework that supports distributed storage and large-scale processing: https://spark.apache.org/. b PARAFAC2 is a flexible variant of CP [START_REF] Harshman | PARAFAC2: Mathematical and technical notes[END_REF]. While the classical CP model requires the tensor factors to be the same for all tensor slices, PARAFAC2 only requires their cross product to be the same and these factors can be different in size slice by slice. c Holt-Winters is an effective time series forecasting procedure [START_REF] Chatfield | The holt-winters forecasting procedure[END_REF].

In stage 1, the authors in [START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF] proposed two solvers for estimating H t and u t , including recursive least-squares (RLS) and simultaneous diagonalization tracking (SDT). Chinh et al.

in [START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF] adopted a well-known subspace tracking algorithm called PETRELS. Dung et al. in [START_REF] Nguyen | Fast adaptive PARAFAC decomposition algorithm with linear complexity[END_REF] applied another subspace tracking algorithm for this task, namely OPAST. In [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF], the same authors also introduced another low-cost tracker to estimate H t with rank-1 updates.

In stage 2, all the existing subspace-based algorithms used the bi-SVD procedure introduced in [START_REF] Strobach | Bi-iteration SVD subspace tracking algorithms[END_REF] to recover U

t (:, i)) ⊤ . Accordingly, the right and left singular vector of the reshaped matrix from H t (:, i) can provide a good estimate of U (1) t (:, i) and U

(2) t (:, i), respectively,

• b i , λ i , a i ← SVD reshape(H t (:, i), [I 2 I 1 ]) • U (1)
t (:, i)← a * i and U

(2)

t (:, i) ← λ i b i
Computation of SVD may be expensive when dealing with large-scale streaming tensors, we can use the alternative update based on power iteration as follows

• H (i) t ← reshape H t (:, i), [J × I] • U (1) t (:, i) ← H (i) t ⊤ U (2)
t-1 (:, i)

• U

(2)

t (:, i) ← H (i) t U (1)
t (:, i)

H (i) t U (1) t (:, i) .
As these algorithms are only designed for tracking third-order streaming tensors, there are still rooms to develop subspace-based methods capable of handling N ≥ 4.

Block-Coordinate Descent

The second approach is based on the block-coordinate descent (BCD) framework in which we decompose the main optimization into two main stages at each time t: (i) estimate the temporal factor U

(N ) t given {U (n) t-1 } N -1
n=1 , and (ii) update the non-temporal factor U

(n) t with 1 ≤ n ≤ N -1 in sequential or parallel given U (N ) t
and the remaining factors. Many tracking algorithms adopt this approach for estimating the low-rank approximation of streaming tensors over time in the literature. We can list here some: TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], OLCP [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF], OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], CP-stream [START_REF] Smith | Streaming tensor factorization for infinite data sources[END_REF],

SPADE [START_REF] Gujral | SPADE: Streaming PARAFAC2 decomposition for large datasets[END_REF], SOFIA [START_REF] Dongjin | Robust factorization of real-world tensor streams with patterns, missing values, and outliers[END_REF], iCP-AM [START_REF] Zeng | Incremental CP tensor decomposition by alternating minimization method[END_REF], ACP [29], and RACP [27]. In what follows, we review their strategy in each stage.

In stage 1, the general formulation of the optimization to estimate the last row u

(N ) t of U (N ) t
can be given by

u (N ) t , O t = argmin u (N ) ,O P t ⊛ Y t -O -U (n) t-1 N -1 n=1 , u (N ) 2 F + ρ u u (N ) 2 2 + ρ O O 1 , (5.12)
where ρ u ∥u∥ 2 2 is for avoiding the ill-posed computation and ρ O ∥O∥ 1 promotes the sparsity in O. Then, the temporal factor U (N ) t is obtained by appending the recent updated u (5.12). In the presence of sparse outliers, (5.12) can be effectively minimized by ADMM or shrinkage-thresholding solvers, as presented in SOFIA [START_REF] Dongjin | Robust factorization of real-world tensor streams with patterns, missing values, and outliers[END_REF] and RACP [27].

In stage 2, the non-temporal factors {U

(n) t } N -1
n=1 can be derived from solving the following optimization

U (n) t = argmin U (n) t τ =1 β t-τ P (n) τ ⊛ U (n) W (n) τ ⊤ + O (n) τ -Y (n) τ 2 F + ρ U R U U (n) , (5.13)
where ρ U R U (.) is a regularization term on U (n) and

W (n) τ =              N -1 i=1,i̸ =n U (i) t-1 ⊙ u ⊤ τ [Jacobi], n-1 i=1 U (i) t ⊙ N -1 i=n+1 U (i) t-1 ⊙ u ⊤ τ [Gauss-Seidel].
(5.14)

Here, we can apply one of the two iterative schemes to update U

(n) t : the Jacobi scheme supports the parallel and/or distributed processing while the Gauss-Seidel scheme is useful for a sequential (serial) one. The regularization can be ∥U

(n) ∥ 2 F for smoothness, ∥U (n) -U (n) t-1 ∥ 2
F for slow timevariation, or U (n) ⪰ 0 for non-negativity constraints. Next, we review two common types of solver for optimizing (5.13): adaptive least-squares filters and stochastic gradient solvers. a) Adaptive Least-Squares (LS) Filters. We can see that the first term of (5.13) is of a weighted LS form very common in adaptive filtering while the second one is to regularize the estimators. Accordingly, (5.13) can be effectively minimized by adaptive LS filters in general and recursive least-squares (RLS) filters in particular.

In [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], Kasai proposed an exponential RLS algorithm called OLSTEC to minimize (5.13) when the observations are outlier-free. OLSTEC is, however, designed for third-order streaming tensors only and its complexities are relatively high compared to other algorithms. Thanh et al. in [29] proposed another RLS algorithm called ACP which is capable of dealing with big streaming tensors of higher order (N ≥ 4). ACP is fast and requires much lower complexity than OLSTEC. Very recently, the same authors in [27] proposed a sliding-window version of ACP robust to both sparse outliers and missing data, namely RACP. Interestingly, three algorithms belong to the class of provable online CP algorithms in which their convergence is guaranteed under certain conditions.

In [START_REF] Vandecappelle | Nonlinear least squares updating of the canonical polyadic decomposition[END_REF], Vandecappelle et al. introduced a nonlinear least-squares (NLS) algorithm for computing the streaming CP decomposition of third-order tensors. In particular, the authors recast the objective function of (5.13) into a truncated exponential window one by incorporating a diagonal weighting matrix L = diag [0, . . . , 0, β L-1 , β L-2 , . . . , β, 1] and then applied a NLS solver to track the tensor factors with time. Following the same line, Smith et al. in [START_REF] Smith | Streaming tensor factorization for infinite data sources[END_REF] proposed 5.4. STREAMING CP DECOMPOSITION another online CP algorithm called CP-stream. This algorithm has the potential to factorize high-order streaming tensors as well as support constraints on streaming CP decomposition such as smoothness and nonnegativity. b) Stochastic Gradient Solvers. Instead of optimizing (5.13) directly, we can minimize its t-th summand:

U (n) t = argmin U (n) P (n) t ⊛ Y (n) t -O (n) t -U (n) W (n) t ⊤ 2 F + ρ U R U U (n) .
(5.15)

Three algorithms TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], OLCP [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF], and SOFIA [START_REF] Dongjin | Robust factorization of real-world tensor streams with patterns, missing values, and outliers[END_REF] adopt this replacement for tracking tensor factors with time. The main difference among them is the type of R U (.). Besides, they obtain different forms of update:

[SOFIA] :

U (n) t = U (n) t-1 + γ t ∆U (n) t , (5.16)
[TeCPSGD] :

U (n) t = 1 - β t tη t U (n) t-1 + 1 η t ∆U (n) t , (5.17)
[OLCP] :

U (n) t = P (n) t Q (n) t -1 with (5.18) P (n) t = P (n) t-1 + ∆P (n) t and Q (n) t = Q (n) t-1 + ∆Q (n) t .
Here, γ t , η t , ∆U

t , ∆P

t , and

∆Q (n) t can be obtained from U (m) t-1 N -1
m=1 and the error

∆Y t = P t ⊛ Y t -O t -{U (n) t-1 } N -1 n=1 , u (N ) t
. It is worth noting that SOFIA is capable of dealing with sparse corruptions. TeCPSGD has the ability to track tensors from missing observations, while OLCP can handle streaming tensors of order greater than 3.

In [START_REF] Zeng | Incremental CP tensor decomposition by alternating minimization method[END_REF], Zeng et al. proposed an incremental ALS algorithm called iCP-AM to minimize a reinforced version of (5.15) which is defined as

U (n) t = argmin U (n) Y (n) t U (n) t-1 U (N ) t-1 ⊙ V (n) t-1 ⊤ -U (n)   u (N ) t Ū(n) t   ⊙ V (n) t ⊤ 2 F , (5.19)
where

V (n) τ = n-1 i=1 U (i) τ ⊙ N -1 i=n+1 U (i)
τ . An appealing feature of iCP-AM against other online CP algorithms is that it has a strategy to deal with the variation of the CP rank over time, i.e., to change the number of low-rank components throughout the tracking process.

In parallel, Gujral et al. in [START_REF] Gujral | SPADE: Streaming PARAFAC2 decomposition for large datasets[END_REF] proposed an online algorithm called SPADE for tracking tensors under the PARAFAC2 format. Specifically, SPADE tracks a fixed (non-temporal) factor along one mode and allows the other tensor factors (modes) to vary with time. Thanks to its stochastic design, SPADE is fast and memory-efficient. However, the stationary assumption that time variation or concept drift is not allowed limits its applicability.

Bayesian Inference

Besides, another good approach for dealing with the problem of streaming CP decomposition is Bayesian inference. The state-of-the-art Bayesian-based streaming CP decomposition algorithms are POST [START_REF] Du | Probabilistic streaming tensor decomposition[END_REF], BRST [START_REF] Zhang | Variational bayesian inference for robust streaming tensor factorization and completion[END_REF], and SBDT [START_REF] Fang | Streaming Bayesian deep tensor factorization[END_REF]. In general, three algorithms start with a prior distribution of unknown parameters and then infer a posterior that best approximates the joint distribution of these parameters on the arrival of new streaming data. The estimated posterior is then used as the prior for the next update. In this subsection, we briefly describe the two online Bayesian inference frameworks which were already used for tensor tracking: (i) streaming variational Bayes (SVB) and (ii) assumed-density filtering (ADF). Also, prior distributions of parameters of interest are reviewed. a) Streaming variational Bayes. The two former algorithms POST and BRST adopted the SVB framework [START_REF] Broderick | Streaming variational bayes[END_REF] which is based on the following Bayes' rule:

p Θ X t-1 ⊞ N Y t p Y t Θ p Θ X t-1 , (5.20)
where Θ denotes the parameters of interest, e.g., tensor factors, CP rank, noise factors, and other parameters. On the arrival of Y t , SVB first uses the current posterior q t-1 (Θ) := p Θ|X t-1 as the prior of Θ, and then integrates with the likelihood of Y t to obtain

pt (Θ) = p Y t Θ q t-1 (Θ), (5.21)
which can be served as an approximation of the joint distribution p(Θ, Y t ) up to a scale factor. The variational posterior q t (Θ) is derived from maximizing the variational model evidence lower bound (ELBO) L(q(Θ)) = E q log pt (Θ)/q(Θ) which is equivalent to minimizing the Kullback-Leibler (KL) divergence:

argmin q KL q(Θ) pt (Θ) = q(Θ) log q(Θ) pt (Θ) dΘ . (5.22)
The optimized form of q t (Θ i ) of (5.22) can be given by

log q t (Θ i ) = E q(Θ/Θ i ) log pt (Θ) + const, (5.23)
where E q(Θ/Θ i ) [.] is an expectation w.r.t. q over all but Θ i .

b) Assumed-Density Filtering. The latter algorithm, SBDT, applied the ADF framework to infer the posterior distribution q t (Θ) over time. Particularly, ADF is an incremental learning framework that allows for computing the approximate posteriors in Bayesian inference for stochastic processes [START_REF] Boyen | Tractable inference for complex stochastic processes[END_REF]. The ADF framework is also grounded on the Bayes' rule (5.20) but 5.4. STREAMING CP DECOMPOSITION utilizes a distribution from the exponential family (e.g., Gaussian distribution) to approximate the current posterior. Instead of minimizing the KL divergence or maximizing the variational ELBO like SVB, ADF projects pt (Θ) into the selected distribution through moment matching to obtain q t (Θ).

c) Prior distributions over Θ. We list common prior distributions over Θ which were already used by POST, BRST, and SBDT.

Prior distribution of tensor factors: All three algorithms assume that the prior over tensor factors is derived from the following Gaussian distribution which is controlled by the hyperpa-

rameter λ = [λ 1 , λ 2 , . . . , λ r ]: p U (n) λ = In i=1 N u (n) i 0, Λ -1 , ∀n ∈ [1, N ], (5.24)
where u

(n) i
is the i-th row of U (n) and Λ = diag(λ) denotes the inverse covariance matrix. Here, λ is supposed to follow a Gamma distribution:

p(λ) = r j=1 Gam λ j |c j , d j , (5.25)
where Gam λ j |c j , d j =

d j c j Γ(c j ) λ j c j -1 e -d j λ j with Γ(z) = ∞ 0 x z-1 e -x dx.
Specifically, the mean and variance of Gam(λ j |c j , d j ) are, respectively, c j /d j and c j /d 2 j which aim to control the magnitude of λ.

Prior distribution of noises: The noise tensor is often assumed to be Gaussian, i.e., N t ∼ i 1 i 2 ...i N N (0, τ -1 ) with a noise precision τ > 0. The parameter τ is further assigned to another Gamma distribution p(τ |a, b) = Gam τ |a, b in the same way as for λ.

Prior distribution of sparse components: Only BRST in [START_REF] Zhang | Variational bayesian inference for robust streaming tensor factorization and completion[END_REF] has the ability to handle sparse outliers. Here, BRST places a Gaussian prior distribution over the sparse O t as

p O t γ = I 1 i 1 I 2 i 2 • • • I N i N N O t i 1 i 2 ...i N 0, γ -1 i 1 i 2 ...i N [Pt] i 1 i 2 ...i N . (5. 26
)
where γ is the sparsity precision parameter. If the value of γ i 1 ...i N is large, the corresponding entry in O t is likely to have a small magnitude. By controlling the value of γ i 1 ...i N , we can control the sparsity of O t .

Prior distribution of NN's weights: SBDT in [START_REF] Fang | Streaming Bayesian deep tensor factorization[END_REF] incorporates neural networks (NN) into tensor factorization. SBDT assigns a spike-and-slab prior distribution over NN weights to sparsify

5.4. STREAMING CP DECOMPOSITION  (3) t U (2) t U (1) t U 1 d 1 1 ( ) [ ] I d r   (1)
ˆt U

ˆt

U ( 3 ) ˆt U 2 2 ( ) [ ] I d r   2 I 2 d 3 I 3 d 3 3 ( ) [ ] I d r   1 I (1) t U (2)
p ω mjt |s mjt = s mjt N ω mjt |0, σ 2 0 + (1 -s mjt )δ(ω mjt ), (5.27)
where δ(.) denotes the delta function and the binary selection indicator s mjt is derived from

p(s mjt ) = Bern(s mjt |ρ 0 ) = ρ s mjt 0 (1 -ρ 0 ) 1-s mjt .

Multi-aspect streaming CP decomposition

In the literature, there are some online algorithms capable of tracking multi-aspect streaming tensors under the CP format, such as MAST [START_REF] Song | Multi-aspect streaming tensor completion[END_REF], OR-MSTC [START_REF] Najafi | Outlier-robust multi-aspect streaming tensor completion and factorization[END_REF], InParTen2 [START_REF] Yang | Multi-aspect incremental tensor decomposition based on distributed in-memory big data systems[END_REF], and DisMASTD [START_REF] Yang | DisMASTD: An efficient distributed multi-aspect streaming tensor decomposition[END_REF]. We refer the readers to Tab. 5.3 for their key features. In what follows, we first describe the main dynamic tensor decomposition (DTD) framework shared by most of these algorithms and then highlight their characteristics in the following text.

For ease of reference, we denote by

X t-1 ∈ R I 1 ו••×I N and X t ∈ R (I 1 +d 1 )ו••×(I N +d N ) the two
successive snapshots at t -1 and t, please see Fig. 5.4 for an illustration. At time t, given X t and the old estimates

{U (n) t-1 } N n=1 of X t-1 , we wish to update {U (n) t } N n=1 such that X t ≈ {U (n) t } N n=1 .
The DTD introduced in [START_REF] Song | Multi-aspect streaming tensor completion[END_REF] offers an online framework for the problem of multi-aspect streaming CP decomposition. Particularly, DTD relaxes the CP representation of X t in the sense that if X t is expressed by {U

(n) t } N n=1 , then its sub-tensor X t-1 can be approximated by { Ū(n) t } N n=1 where Ū(n) t ∈ R In×r is the sub-matrix of U (n) t ∈ R (In+d)
{U (n) } N n=1 ℓ Y t , U (n) N n=1 + ρ N n=1 U (n) * , (5.28)
where the loss function ℓ(.) is defined as

ℓ Y t , U (n) N n=1 = µ {U (n) t-1 } N n=1 -{ Ū(n) } N n=1 2 F + P Ωt Y t -P Ωt {U (n) } N n=1 2 F . (5.29)
Here, Ω t denotes the set of observed entries and µ, ρ > 0 are two regularized parameters. Depending on the type of constraints, additional information imposed and the method of optimization, we can obtain several types of estimators for tracking multi-aspect streaming tensors with time under the DTD framework.

In [START_REF] Song | Multi-aspect streaming tensor completion[END_REF], Song et al. developed the so-called MAST algorithm for tracking multi-aspect streaming tensors. The authors recast (5.28) into a constrained minimization and then formed the following Lagrangian function

L Y t , Θ = N n=1 ρ Z (n) * + Λ (n) , Z (n) -U (n) + η 2 Z (n) -U (n) 2 F + ℓ Y t , U (n) N n=1 , (5.30)
where Θ = {U (n) , Z (n) , Λ (n) } N n=1 with auxiliary matrices {Z (n) } N n=1 and Lagrange multiplier matrices {Λ (n) } N n=1 , and η > 0 is a regularization parameter. Since terms of (5.30) are all convex, it can be effectively minimized by several methods. In particular, MAST applies an ADMM solver to minimize (5.30) in order to balance the trade-off between effectiveness and efficiency in tracking process.

Since MAST is not designed for handling sparse outliers, Najafi et al. in [START_REF] Najafi | Outlier-robust multi-aspect streaming tensor completion and factorization[END_REF] introduced a robust version of MAST called OR-MSTC. In the presence of sparse outliers, the authors proposed to regularize the objective function of (5.28) by adding an ℓ 1 -norm regularization term λ∥O∥ 1 and replacing Y t with Y t -O in the first term of ℓ(.) in (5.30). Because the term λ∥O∥ 1 is convex, OR-MSTC also adopts the well-known ADMM method in a similar way to MAST.

In [START_REF] Yang | Multi-aspect incremental tensor decomposition based on distributed in-memory big data systems[END_REF], Yang et al. proposed a distributed version of MAST called InParTen2. Thanks to Apache Spark, it can handle large-scale streaming tensors efficiently with a limited memory.

However, the use of InParTen2 is limited for third-order streaming tensors only. In [START_REF] Yang | DisMASTD: An efficient distributed multi-aspect streaming tensor decomposition[END_REF], Yang et al. introduced another distributed method called DisMASTD capable of dealing with tensors of higher order. One of appealing feature of DisMASTD is that it can avoid repetitive computation and reduce network communication cost. (2017 [START_REF] Song | Multi-aspect streaming tensor completion[END_REF]) (2019 [START_REF] Najafi | Outlier-robust multi-aspect streaming tensor completion and factorization[END_REF]) (2020 [START_REF] Yang | Multi-aspect incremental tensor decomposition based on distributed in-memory big data systems[END_REF]) (2021 [START_REF] Yang | DisMASTD: An efficient distributed multi-aspect streaming tensor decomposition[END_REF])

STREAMING TUCKER DECOMPOSITION

Missing? ✓ ✓ ✗ ✗ Outliers? ✗ ✓ ✗ ✗
High-order?

(N ≥ 4) ✓ ✓ ✗ ✓ Distributed? ✗ ✗ ✓ ✓ (1) t U 1 [] Ir  2 [] Ir  New Observations 1 t  X (2) t U (1) t U  (1 ) t U 11 [] Ir  2 2 [ ] Ir  1 t  G (2) 1 t  U (1) 1 t  U  1 t  X 1 t  X 2 I 1 I 3 I 1  2 I 1 I 3 I 12 ( 1) [ ] r r t    1 2 1 [ ] rr  t G 1 t  G t Y 1 2 [ ] r r t  

Streaming Tucker Decomposition

In the literature, there are many online tensor methods proposed for factorizing streaming tensors. We can broadly categorize them into three main classes: (i) online tensor dictionary learning, (ii) tensor subspace tracking, and (iii) multi-aspect streaming Tucker decomposition. Specifically, the first two classes are designed for two specific cases of single-aspect streaming Tucker decompositions, while the latter class is for multi-aspect streaming tensors.

Online Tensor Dictionary Learning

In the class of online tensor dictionary learning methods, we are particularly interested in a specific case of single-aspect streaming Tucker decomposition where the underlying tensor [START_REF] Kasai | Low-rank tensor completion: A Riemannian manifold preconditioning approach[END_REF] manifold

X T ∈ R I 1 ו••×I N -1 ×T -
HO-RLSL [245] ✗ ✓ ✓ ✗ 3I 2 O I 3 -For N = 4 only IHOSVD [246] ✗ ✗ ✓ ✗ O N (I/d) 2(N -1)
-Supports distributed computing d: number of cores -Adopts RoundRobin process + columnwise Jacobi-rotation

MIHOSVD [247] ✗ ✗ ✓ ✗ O N (I/d) 2(N -1)
-Supports distributed computing d: number of cores -Adopts tree-based integration + columnwise Jacobi-rotation

SIITA [248] ✓ ✗ ✓ ✗ O K(r N |Ω| + N IM r) -Multi-
✗ ✓ ✓ ✗ O K(rI N -1 + I 2 r N -1 ) -Applies threshold-based outlier K: iterations detection + L1-HOOI BASS-Tucker ✗ ✗ ✓ ✗ O r 3(N -1) + (Ir) N -1 +N r 3 I N -1
-Sparse decomposition [START_REF] Fang | Bayesian streaming sparse Tucker decomposition[END_REF] -Uses Bayesian inference + ADF [START_REF] Jang | Fast and memory-efficient tucker decomposition for answering diverse time range queries[END_REF] temporal slices -Requires a preprocessing phase

SBDT [229] ✗ ✗ ✓ ✗ O N Ir + KI N -1 -Uses Bayesian inference + ADF K: number of weights in NNs -Incorporates NNs Zoom-Tucker ✗ ✗ ✓ ✗ O KBN rI N -1 + KN 2 r N +1 +KN 2 r 2 I) K: iterations & B : blocks -Supports multiple coming
RI/BK-NTD ✗ ✗ ✓ ✗ O KN (Ir) N -Nonnegative decomposition [257] K: iterations -Uses NNLS + BCD ATD [29] ✓ ✗ ✓ ✓ O r|Ω| + r 2N |S1| + r 2 |S2| +r 2 I N -2 with |S1|, |S2| : size of sampling sets -Uses BCD + Sampling -Supports parallel computing ⋆ Suppose that I1 = I2 = • • • = IN = I, r1 = r2 = • • • = rN = r,
and |Ω| is the number of observed elements.

# Abbreviations: ISVD, (incremental SVD), SGD (stochastic gradient descent), MGS (modified Gram-Schmidt process), BCD (block-coordinate descent), ADF (assume-density filtering), NN (neural network), and NNLS (nonnegative constrained least-squares solver).

a Pseudo inputs: a small active pseudo set, which is not necessarily required to be a subset of the real data, is introduced to break the dependencies between outputs and hence avoid the explicit computation of the full covariance matrix.

• • • × I N -1 -is supposed to be modelled by

X T = G T ; U (n) N -1 n=1 , I T , (5.31)
where the core tensor G T is of size r 1 × • • • × r N -1 × T (i.e., r N = T ), the tensor factors

{U (n) } N -1 n=1 , U (n) ∈ R
In×rn are of fixed size, and the last factor U (N ) is an identify matrix. Specifically, the t-th temporal slice Y t of X T is expressed as

Y t = G t ; U (n) N -1 n=1 , t = 1, 2, . . . , T, (5.32)
where G t ∈ R r 1 ×r 2 ו••×r N -1 is the t-th slice of the core tensor G T . The primary objective here is to estimate G t and incrementally update {U (n) } N -1 n=1 on the arrival of Y t at each time t. In what follows, we review two main approaches to deal with this problem. a) Incremental Subspace Learning on Tensor Unfolding Matrices. A natural and very first approach for streaming Tucker decomposition is to incrementally update the subspaces covering unfolding matrices of the underlying tensor. The central idea of this approach stems from the fact that the n-th tensor factor U (n) t which is derived from the standard HOSVD is given by

U (n) t = EVD X (n) t-1 , Y (n) t X (n) t-1 , Y (n) t ⊤ , (5.33)
where

X (n) t-1 = Y (n) 1 , . . . , Y (n) t-1 with Y (n) τ
is the mode-n unfolding matrix of Y τ . Accordingly at time t, we can apply the following dynamic tensor analysis (DTA) framework introduced in [START_REF] Sun | Beyond streams and graphs: Dynamic tensor analysis[END_REF][START_REF] Sun | Incremental tensor analysis: Theory and applications[END_REF] to estimate G t and update {U

(n) t } N -1 n=1 : C (n) t ← βC (n) t-1 + Y (n) t ⊤ Y (n) t , (5.34a)
U (n) t ← eig C (n) t , r , (5.34b)
G t ← Y t , (U (n) t ) ⊤ N -1 n=1 , (5.34c)
where 0 < β ≤ 1 is a forgetting factor and eig(C

(n) t , r) computes the top r principal eigenvectors of C (n)
t . Since the two steps (5.34a) and (5.34b) are generally expensive, there have been some studies offering good modifications or fast alternatives for (5.34).

In [START_REF] Sun | Beyond streams and graphs: Dynamic tensor analysis[END_REF][START_REF] Sun | Incremental tensor analysis: Theory and applications[END_REF] n=1 . Specifically, instead of computing (5.34a), IRTSA applies a fast incremental SVD (ISVD) proposed by Ross et al. in [START_REF] Ross | Incremental learning for robust visual tracking[END_REF] on the mode-n unfolding matrix X (5.33). Thanks to ISVD, IRTSA shares the same order of computational complexity with STA while offers a better estimation than STA for the problem of background modelling and object tracking. Although the current version of IRTSA is designed for factorizing threeorder streaming tensors, it is not difficult to extend IRTSA for dealing with higher-order tensors.

(n) t = X (n) t-1 , Y (n) t in
Besides, a modified version of IRTSA was introduced by Zang et al. in [START_REF] Zhang | An incremental tensor factorization approach for web service recommendation[END_REF] for the problem of web service recommendation.

In [START_REF] Kuang | A tensor-based approach for big data representation and dimensionality reduction[END_REF], Kuang et al. also proposed an incremental SVD-based streaming Tucker decomposition, namely IHOSVD. In particular, this algorithm performs the following three processes in a serial manner: (i) applies a recursive SVD method to compute singular values and singular vectors of unfolding matrices of the new tensor, (ii) merges the new results with the old estimations from past observations, and (iii) obtains the core tensor with n-mode products. Theoretical analyses and experimental results on intelligent transportation applications demonstrate the effectiveness of IHOSVD.

In [START_REF] Li | Robust tensor subspace learning for anomaly detection[END_REF], Li et al. modified slightly the recursive update of the covariance matrix C (n) t in (5.34a) as follows

C (n) t = (1 -α)C (n) t-1 + α Y (n) t ⊤ Y (n) t , (5.35)
with a weight 1 ≥ α > 0 and then introduced a robust incremental algorithm called RTSL which has the potential to model background and detect anomalies in applications of computer vision.

Since RTSL still applies directly the DTA framework, its complexity is relatively high. Thus, it may become inefficient for handling large-scale and high dimensional streaming data.

Some other algorithms for streaming Tucker decomposition belonging to this group were presented in [START_REF] Ozdemir | Recursive tensor subspace tracking for dynamic brain network analysis[END_REF][START_REF] Wang | A distributed HOSVD method with its incremental computation for big data in cyber-physical-social systems[END_REF][START_REF] Yang | A multi-order distributed HOSVD with its incremental computing for big services in cyberphysical-social systems[END_REF][START_REF] Wang | Improved multi-order distributed HOSVD with its incremental computing for smart city services[END_REF], focusing on specific applications such as dynamic brain network analysis, smart city services, cyber-physical-social networks and systems.

b) Online Multimodal Dictionary Learning. Another good strategy for the problem of single-aspect tensor tracking is to apply online multimodal dictionary learning (OMDL) techniques. As OMDL is a stochastic version of the multimodal dictionary (multilinear subspace) learning [START_REF] Lu | A survey of multilinear subspace learning for tensor data[END_REF], it allows estimating dictionaries (i.e., tensor factors) with one-pass processing.

In the literature, there exist some algorithms applying OMDL for tracking the low multilinearrank component of streaming tensors with time, such as OTDL [START_REF] Traoré | Online multimodal dictionary learning[END_REF], ODL [START_REF] Zhao | Learning separable dictionaries for sparse tensor representation: An online approach[END_REF], ORLTM [START_REF] Li | Online robust low-rank tensor modeling for streaming data analysis[END_REF],

OLRTR [START_REF] Hu | Streaming data preprocessing via online tensor recovery for large environmental sensor networks[END_REF], and D-L1-Tucker [START_REF] Chachlakis | Dynamic L1-norm Tucker tensor decomposition[END_REF].

The two former algorithms OTDL and ODL adopt the typical two-step learning procedure to track the tensor factors over time, namely (i) tensor coding or inference of coefficients in the core tensor and (ii) dictionary update per each tensor mode.

STREAMING TUCKER DECOMPOSITION

Step 1: Tensor Coding. When Y t is observed, the general formulation of optimization for this step is given by:

G t = argmin G Y t -G; U (n) t-1 N -1 n=1 2 F + ρ G R G (G) , (5.36)
where ρ G R G (.) is a regularization term on the core tensor G to promote sparsity or nonnegativity for instance. Since the first term of (5.36) is differentiable while the second term may admit a proximal operator (e.g., ℓ p -norm), OTDL and ODL applied proximal methods to minimize it.

Step 2: Dictionary Update. When G t is estimated, the BCD framework can be used to update

U (n)
t . Specifically, both algorithms optimize the following minimization:

U (n) t = argmin U (n) t τ =1 Y τ -G τ ; U (n) t-1 N -1 n=1 2 F + ρ U R U U (n) , (5.37)
with a penalty term ρ U R U (.) on U (n) . Interestingly, (5.37) can be recast into the standard leastsquares cost function which is very common in adaptive filtering theory. Accordingly, OTDL introduced an effective recursive least-squares (RLS) solver to optimize it. Meanwhile, ODL used the stochastic gradient descent method to estimate U (n) t

with a low cost.

The next two algorithms ORLTM and OLRTR, on the other hand, estimated the tensor factors without the need of tensor coding. In particular, the tensor factor U (n) is directly derived from the following optimization

U (n) t = argmin U (n) t τ =1 ℓ Y τ , U (n) + ρ U R U U (n) , (5.38)
where the loss function ℓ(.) is defined as

ℓ Y τ , U (n) = min R (n) ,E (n) Y (n) τ -U (n) R (n) -O (n) 2 F + λ 1 E (n) 1 + λ 2 R R (R (n) ) . (5.39)
Here, R (n) and O (n) play the role of the coefficient and error, respectively. The main difference between ORLTM and OLRTR is the type of R R (.) used. Specifically, OLRTR uses the simple

Frobenius norm regularization R R (R (n) ) = ∥R (n) ∥ 2 F , while ORLTM reinforces R (n) = W (n) Z (n) and then forms R R (R (n) ) = ∥W (n) ∥ 2 F + ∥Z (n) ∥ 2 F .
Intuitively, the minimization (5.38) may be regarded as a robust version of (5.37) which aims to deal with sparse corruptions. Also, the minimization (5.39) is not difficult to solve since its terms are all convex. Hence, both OLRTR and ORLTM applied the RLS method to update U ]

Ir  3 ( 1)
[]

Ir  New Observations t G (3) t U 1 t  X (2) t U (1) t U  (1 ) t U 1 1 [ ] I r  2 2 [ ] I r  1 t  G (3) 1 t  U (2) 1 t  U (1) 1 t  U  3 3 [ ] I r  (3) t u 1 t  X 2 I 1 I 3 I 1  (3) t U 2 I 1 I 3 I 1 2 3
[] r r r  arrival of Y t , D-L1-Tucker first identifies whether Y t is an anomaly or not based on its reliability which is defined as

r t = Y t ; (U (n) t-1 ) ⊤ N -1 n=1 2 F Y t -2 F .
(5.40)

If r t ≤ τ where τ ∈ [0, 1] is a predefined threshold, Y t is labelled as an outlier slice and then it is disregarded. Otherwise, Y t is considered as reliable and useful for tracking process. In such a case, D-L1-Tucker appends Y t to the memory set Z t = Z t-1 ∪ Y t and then applies the batch L1-HOOI algorithm proposed in [START_REF] Chachlakis | L1-norm Tucker tensor decomposition[END_REF] for factorizing Z t in order to obtain tensor factors. After that, Z t is re-updated by removing the oldest measurement for the next processing. D-L1-Tucker requires a good batch initialization and its tracking ability is dependent on the threshold τ and the memory size M to store Z t .

Tensor Subspace Tracking

Apart from the model (5.31), the tensor

X T ∈ R I 1 ו••×I N -1
×T and its t-th temporal slice Y t with 1 ≤ t ≤ T can be modelled as follows

X T = G; U (n) N n=1 , (5.41)
Y t = G; U (n) N -1 n=1 , u (N ) t , (5.42)
where the core tensor G ∈ R r 1 ×r 2 ו••×r N and {U (n) } N -1 n=1 with U (n) ∈ R In×rn are of fixed size except the last factor U (N ) ∈ R T ×r N , and u (N ) t ∈ R 1×r N is the t-th row of U (N ) , see Fig. 5.6 for an illustration. At each time t, given old estimations G t-1 and {U

(n) t-1 } N -1 n=1 , we are interested in tracking G t , u (N ) t and {U (n) t } N -1
n=1 which can compactly represent the temporal slice Y t . We 5.5. STREAMING TUCKER DECOMPOSITION refer this problem to as tensor subspace tracking. 1 It is worth mentioning that single-aspect streaming CP methods also belong to this class as the core tensor G is constrained to be identity. In the literature, there exist some tensor subspace tracking methods which have the potential to deal with a general case of G. Each method adopts a different strategy to factorize streaming tensors. In what follows, we briefly describe their main features in chronological order.

M r = X ∈ R I 1 ×I 2 ו••×I N | rank(X ) := r = [r 1 , r 2 , . . . , r N ] ,
Riemannian optimization can offer a good approach for tensor decomposition and completion [START_REF] Kressner | Low-rank tensor completion by Riemannian optimization[END_REF]. Accordingly, the authors proposed an efficient Riemannian gradient based method to estimate the low multilinear-rank component of tensors. The proposed method consists of a rank-one Riemannian gradient computation and a retraction step. Specifically, a novel Riemannian metric on the tangent space of M r and its quotient manifold was introduced to enable the Riemannian optimization framework. Furthermore, a map that combines all retractions on the individual manifolds of tensor factors was used to transform the estimations to the tensor manifold.

1 This name stems from the following observation: we can recast (5.42) into the standard form

yt = Dut, (5.43)
where yt = vec(Y t ), ut = u (N ) t ⊤ and D is the transpose of the mode-N unfolding matrix of G; {U (n) } N -1 n=1 . Intuitively, (5.43) may be regarded as the data model which is very common and widely used in the problem of subspace tracking where we wish to incrementally update D on the arrival of yt at each time t. Since the subspace matrix D has a tensor structure, we label this problem as "tensor subspace tracking" without hesitation. (SVB) framework to infer the posterior of parameters of interest (e.g., tensor core, tensor factors, and nuisance parameters) over time. In addition, BASS-Tucker also utilizes the same priors for the tensor factors and noise variance except that of the core tensor. Here, the following spike-and-slab prior is used to model the core tensor:

p S|ρ 0 = r 1 j 1 =1 r 2 j 2 =1 • • • r N j N =1 Bern s j 1 j 2 ...j N |ρ 0 , (5.44)
p(G|S) = r 1 j 1 =1 r 2 j 2 =1 • • • r N j N =1 s j 1 j 2 ...j N N g j 1 j 2 ...j N |0, σ 2 0 + (1 -s j 1 j 2 ...j N )δ(g j 1 j 2 ...j N ), (5.45)
where S ∈ R r 1 ×r 2 ו••×r N is a binary tensor, Bern(.|ρ 0 ) is the Bernoulli distribution with probability ρ 0 , and δ(.) is the Delta function. We refer the readers to subsection 5.4.3 for details on prior distributions of {U (n) } N -1 n=1 and other model parameters as well as how the SVB framework works. d) Block-Coordinate Descent. There are three online Tucker algorithms using the BCD framework, including ATD [29], RT-NTD [START_REF] Zdunek | Incremental nonnegative Tucker decomposition with blockcoordinate descent and recursive approaches[END_REF] and BK-NTD [START_REF] Zdunek | Incremental nonnegative Tucker decomposition with blockcoordinate descent and recursive approaches[END_REF]. In general, they go through the following stages when Y t arrives: given old estimations G t-1 and {U

(n) t-1 } N -1 n=1 . Generally, u (N ) t can be derived from u (N ) t = argmin u (N ) Y t -G t-1 ; {U (n) t-1 } N -1 n=1 , u (N ) 2 F + ρ u R u (u (N ) ) . ( 5
t-1 and the remaining factors, 1 ≤ n ≤ N -1. The main optimization can be given by

U (n) t = argmin U (n) t τ =1 β t-τ ℓ(Y τ , U (n) ) + ρ U R U U (n) , (5.47)
where

ℓ(Y τ , U (n) ) = Y (n) τ -U (n) W (n) τ 2 F , Y (n)
τ and W

(n)

τ are respectively the mode-n unfolding matrices of Y τ and W τ . Here, the coefficient tensor W τ is defined as

W τ = G t-1 ; {U (m) t-1 } N -1 m=1,m̸ =n , u (N ) τ .
(5.48)

 (3) t U (2) t U (1) t U 1 d 1 1 1 () [ ] I d r  (1)
ˆt U

I d r   1 I (1) t U (2) t U ( 3 ) t U (0,0,0) t X New Observations 1 I 1 d  3 I  3 d 2 I 2 d  (0,1,0) t X (1,0,0) t X (1,1,0) t X (0,1,1) t X (1,1,1) t X (1,0,1) t X (0,0,1) t X (0,0,0) 1 t t   X X 1 2 3 [ ]
G t = argmin G t τ =1 β t-τ X (1) τ -U (1) t G (1) Z τ 2 F + ρ G R G (G) , (5.49)
where (.) (1) denotes the mode-1 unfolding matrix and

Z τ = u τ ⊗ N n=2 U (n) t .
Here, R u (.), R U (.), and R G (.) are regularization terms on the coefficient u

(N )
t , the factor U (n) t , and the core tensor G t , respectively. These penalties can be nonnegativity, smoothness, or sparsity depending on the specific application.

The former ATD algorithm was proposed by Thanh et al. in [29] which is capable of tracking the low multilinear-rank approximation of streaming tensors from highly incomplete observations. In stage 1, ATD particularly recasts (5.46) into a standard LS optimization and then applies a randomized LS technique to minimize it. In stage 2, ATD introduces a recursive LS solver to optimize (5.47) in an efficient way. Instead of solving (5.49) directly, ATD applies the stochastic gradient descent to obtain its solution.

The two latter RI-NTD and BK-NTD algorithms were proposed by Zdunek et al. in [START_REF] Zdunek | Incremental nonnegative Tucker decomposition with blockcoordinate descent and recursive approaches[END_REF] for factorizing nonnegative tensors from streaming data. Both algorithms perform nonnegative leastsquare (NNLS) solvers to incrementally update the tensor factors and the core tensor. Particularly, RI-NTD utilizes a recursive strategy involving the nonnegatively constrained Gauss-Seidel method while BK-NTD adopts the block Kaczmarz method. Similar to ATD, both RI-NTD and BK-NTD estimate the core tensor using only the new coming data via a stochastic optimization.

Multi-aspect streaming Tucker decomposition

Besides single-aspect streaming Tucker decomposition methods, few online techniques are capable of tracking multi-aspect streaming tensors under the Tucker format over time, such as SITTA in [START_REF] Madhav | Inductive framework for multiaspect streaming tensor completion with side information[END_REF] and eOTD in [START_REF] Xiao | eOTD: An efficient online tucker decomposition for higher order tensors[END_REF].

SIITA in [START_REF] Madhav | Inductive framework for multiaspect streaming tensor completion with side information[END_REF] offers an online inductive framework for tracking the low-rank tensor approximation of multi-aspect streaming tensors as well as completing their missing data with side information. On the arrival of new data, SIITA particularly minimizes the following optimization

argmin G,{U (n) ,A (n) } N n=1 f t Y t , {S (n) t } N n=1 , G, {U (n) } N n=1 , (5.50)
with

f t Y t , {S (n) } N n=1 , G, {U (n) } N n=1 = P Ωt Y t -P Ωt G; S (n) t U (n) N n=1 2 F + ρ G G 2 F + N n=1 ρ n U (n) 2 F , (5.51)
where {S

(n) t } N n=1 with S t ∈ R Mn×In is the set of side information matrices and ρ G , {ρ i } N i=1 are regularization parameters. Here, SIITA incorporates the side information into the data model by using {S (n) t } N n=1 as multiplicative terms. Accordingly, SIITA can accelerate the tracking process because the product S (n) t U (n) transforms the dimensionality of variables from I n to M n , and typically with M n ≪ I n . As every term of (5.50) are convex, SITTA adopts the gradient descent to minimize it. Besides, a simple variant of SIITA namely NN-SITTA was also obtained for nonnegative tensor decomposition. NN-SITTA is specifically derived from projecting the estimates of SIITA into their nonnegative orthant at each time t.

In [START_REF] Xiao | eOTD: An efficient online tucker decomposition for higher order tensors[END_REF], Xiao et al. proposed the so-called eOTD algorithm for the multi-aspect tensor tracking problem. Unlike SIITA, eOTD adopts the divide and conquer paradigm to deal with multi-aspect streaming tensors. In particular, it divides the underlying tensor X t into 2 N subtensors X (i 1 ,...,i N ) t with i n ∈ {0, 1}, 1 ≤ n ≤ N , and X (0,...,0) t = X t-1 , see Fig. 5.7 for an illustration. These sub-tensors are grouped into N classes {X n } N n=1 based on the sum of subindices. For example, for a third-order tensor, we have

X 1 = X (1,0,0) t , X (0,1,0) t , X (0,0,1) t , X 2 = X (1,1,0) t , X (1,0,1) t , X (0,1,1) t
, and

X 3 = {X (1,1,1) t }. If a sub-tensor X (i 1 ,...,i N ) t ∈ G n , factorizing it will results in X (i 1 ,...,i N ) t = G t , {V (n) t } N n=1 where V (n) t = Û(n) t if i n = 1 and V (n) t = U (n) t if i n = 0.
Here, the matrix Û(n) t is constantly updated as follows

Û(n) new = α Û(n) old + (1 -α)X t (i 1 ,...,i N ) n G (n) in # .
(5.52)

The tensor factor where the modified Gram-Schmidt process was applied to compute the orth(.) operation. Finally, the tensor core G t of fixed size is estimated by

U (n) t is specifically derived from U (n) t = orth [U (n) t-1 ; Û(n) new ] = [ Ū(n) 1 ; Û(n) t ] 5.6. OTHER STREAMING TENSOR DECOMPOSITIONS  1 1  1 2  1 N  (1) t G (2) t G ( 1) N t  G () 1 N t  G 1 1 N    1 t  X t Y 11 [] Ir  1 2 2 [] r I r  2 1 1 [ ] N N N r I r     1 [ ] N r t   ( ) N t G t X
G t = G t-1 , Ū(n) t ⊤ U (n) t-1 N n=1 + (i 1 ,...,i N )̸ =(0,...,0) X (i 1 ,...,i N ) t , Û(n) t N n=1 .
(5.53)

An appealing feature of eOTD is that throughout the tracking process, eOTD only uses cheap tensor-matrix multiplications and pseudo-inverse operations instead of computing the expensive SVDs on big matrices. This makes eOTD easy for applying to large-scale applications.

Other Streaming Tensor Decompositions

Apart from the two most popular streaming CP and Tucker decompositions, some online methods are capable of tracking tensors under other multiway models. This section focuses on tracking algorithms that exploit TT, BTD, and t-SVD formats to construct the low-rank tensor approximation in the streaming model.

Streaming Tensor-Train Decomposition

Despite success in the batch setting, TT decomposition has not gained in popularity as CP and Tucker for tensor tracking. In the literature, there exist few tracking algorithms developed for the problem of single-aspect tensor tracking under the TT format, see Fig. 5.8 for an illustration.

In [30,32,35], Thanh et al. proposed three adaptive TT algorithms called TT-FOA, ATT, and ROBOT for factorizing tensors in an online fashion. Particularly, TT-FOA in [32] is, to the best of our knowledge, the very first of its kind in the literature. However, its practical use is limited due to the lack of robustness to data corruption. To overcome the drawback, ATT in [30] and ROBOT in [35] were developed to deal with missing data and sparse outliers, respectively.

All three algorithms share the same optimization framework where block-coordinate gradient 5.6. OTHER STREAMING TENSOR DECOMPOSITIONS (BCD) and recursive least-squares (RLS) methods are utilized to minimize the cost function. In particular, a general formulation of the optimization problems can be written as

{G (n) t } N n=1 , O t = argmin {G (n) } N n=1 ,O t τ =1 β t-τ P τ ⊛ G (1) × 1 2 • • • × 1 N -1 G (N -1) × 1 N G (N ) τ + O τ -Y τ 2 F + ρ O R O O τ + ρ G R G G (n) N -1 n=1 , (5.54)
where β ∈ (0, 1] is a forgetting factor to reduce the impact of old observations; R O (O τ ) and

R G {G (n) } N -1 n=1
are two regularization terms. Specifically, TT-FOA does not impose the two

penalties; ATT adopts R G {G (n) } N -1 n=1 = N -1 n=1 G (n) -G (n) t-1 2
F to control the smoothness of TT-cores over time; and ROBOT applies the

ℓ 1 -norm regularization R O (O τ ) = ∥O τ ∥ 1 to promote the sparsity on O τ .
Thanks to the BCD framework, (7.44)

(n) t } N -1 n=1 when G (N ) t
and O t (if any) are given in stage 1.

In parallel, Liu et al. in [START_REF] Liu | An incremental tensor-train decomposition for cyber-physical-social big data[END_REF] proposed an incremental TT method called iTTD to factorize tensors having one temporal mode. Specifically, iTTD considers coming data streams as individual tensors and then factorizes them into TT-cores. The results are appended to old estimates derived from past observations. In [START_REF] Wang | ADTT: A highly efficient distributed tensor-train decomposition method for IIoT big data[END_REF], Wang et al. also developed an incremental TT method called AITT to decompose tensors from industrial IoT data streams. By exploiting a relationship between the directly reshaped matrix and integration of tensor unfolding matrices, AITT can estimate effectively the underlying TT-cores. However, the two frameworks of iTTD and AITT are not really online streaming learning ones but incremental batch learning. Therefore, they are not useful for data streams from dynamical observations in time-varying environments.

Streaming Block-Term Decomposition

The block-term decomposition (BTD) unifies the two well-known CP and Tucker decompositions, and thus, the tracking algorithms under the CP and Tucker formats principally belong to the class of the streaming BTD with one block. When the number of blocks is greater than 2, there are only two BTD methods able to deal with streaming tensors, including OnlineBTD [START_REF] Gujral | OnlineBTD: Streaming algorithms to track the block term decomposition of large tensors[END_REF] and O-BTD-RLS [START_REF] Rontogiannis | Online rank-revealing block-term tensor decomposition[END_REF].

The former method was proposed by Gujral et al. in [START_REF] Gujral | OnlineBTD: Streaming algorithms to track the block term decomposition of large tensors[END_REF] for tracking tensors under the generalized BTD format of L blocks and a multilinear rank-(r 1 , r 2 , . . . , r N ). On the arrival of  ... the temporal slice Y t , OnlineBTD performs the following minimization:

t X 1  t X t Y   ) 1 ( 1 U ) 2 ( 1 U ) 1 ( L U ) 2 ( L U 1 u L u
argmin {G i } r i=1 ,{U (n) } N n=1 Y t - r i=1 G i , {U (n) i } N n=1 2 F , (5.55)
where

U (n) = U (n) 1 , U (n) 2 , . . . , U (n) r with U (n) i ∈ R In×rn and G i ∈ R r 1 ×r 2 ו••×r N , 1 ≤ n ≤ N, 1 ≤ i ≤ r.
Here, {U (n) } N -1 n=1 are supposed to remain unchanged with time except the last tensor factor U (N ) . Prior information of L and rank-(r 1 , r 2 , . . . , r N ) are known in advance. Old estimates of the core tensors and tensor factors of X t-1 are used as a "warm start" for OnlineBTD at each time t. To speed up the tracking, OnlineBTD utilizes (i) an accelerated matricized tensor times Kronecker product, (ii) the pseudo-inverse operator using LU decomposition, and (iii) a dynamic programming strategy introduced by Zhou et al. in [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF] to avoid the re-computation of duplicated Kronecker products.

The second method was introduced by Rontogiannis et al. in [START_REF] Rontogiannis | Online rank-revealing block-term tensor decomposition[END_REF]. Specifically, O-BTD-RLS is designed for tracking the low rank-(r, r, 1) terms of three-order streaming tensors (i.e., r 1 = r 2 = r and r 3 = 1), see Fig. 5.9 for an illustration. In particular, the tensor factors of the underlying tensor are incrementally updated by minimizing the following objective function:

argmin {U (n) } 3 n=1 t τ =1 β t-τ Y τ -U (1) W τ U (2) ⊤ 2 F + ρ 1 Ξu l 2 2 + η 2 + ρ 2 L l=1 r k=1 u (1) l,k 2 2 + u (2) l,k 2 2 + η 2 , (5.56)
Here,

U (n) = U (n) 1 , U (n) 2 , . . . , U (n) L with U (n) l
∈ R In×r is the n-th tensor factor of interest, and

u (n) l,k is the k-th column of U (n)
l , n = 1, 2; u τ and u l are the τ -th row and l-th column of the 5.6. OTHER STREAMING TENSOR DECOMPOSITIONS temporal factor U (3) ∈ R t×L , respectively; W τ = diag(u τ ) ⊗ I r and Ξ = diag(β t-1 , . . . , β, 1); ρ 1 and ρ 2 are two regularization parameters; and η 2 is a small positive number to promote smoothness at zero. Here, the former term of (5.56) has the form of weighted least-squares while two latter terms are regularizations. Accordingly, an efficient recursive least-squares solver was introduced to minimize (5.56) effectively. An appealing feature of O-BTD-RLS is that it has the ability to reveal the BTD ranks over time by specifying the number of columns of the tensor factors which are non-negligible in magnitude at each time t.

Streaming t-SVD Decomposition

Similar to TT and BTD, streaming t-SVD is still in its early stage. In the literature, there exists only two works of Zhang et al. in [START_REF] Zhang | An online tensor robust PCA algorithm for sequential 2D data[END_REF] and Gilman et al. in [START_REF] Gilman | Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD[END_REF][START_REF] Gilman | Online tensor completion and free submodule tracking with the t-SVD[END_REF] addressing the problem of tensor tracking under the t-SVD format.

In [START_REF] Zhang | An online tensor robust PCA algorithm for sequential 2D data[END_REF], Zhang et al. introduced an online tensor PCA for sequential 2D data based on the t-SVD structure. When Y t arrives, the proposed algorithm updates:

• The coefficient matrix W t and the sparse outlier O t from solving the following minimization

{W t , O t } = argmin W,O 1 2 Y t -U t-1 * W -O 2 F + λ 1 2 ∥W∥ 2 F + λ 2 ∥O∥ 1 . (5.57)
• The low tubal-rank tensor U t (a.k.a. basis dictionary) from taking iFFT of the tensor Û t along the third dimension where Û t is specifically derived from

Ût = argmin Û 1 2 tr Û⊤ Ât + I 3 λ 1 I Û -tr Û⊤ Bt . (5.58)
Here,

Ât = diag(FFT(A t )) with A t = A t-1 + W t * W ⊤ t , Bt = diag(FFT(B t )) with B t = B t-1 + (Y t -O t ) * W ⊤
t , and the solution Ût is a matricization of Û t .

As the online tensor PCA above is not designed for handling missing data, Gilman et al.

in [START_REF] Gilman | Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD[END_REF][START_REF] Gilman | Online tensor completion and free submodule tracking with the t-SVD[END_REF] proposed another algorithm called TOUCAN which is capable of tracking tensors from missing observations. Specifically, the authors proposed to solve the constrained minimization

{U t , w t } = argmin U,w t τ =1 F Ωτ y τ -Uw τ 2 2 subject to U ⊤ U = I rI 3 , (5.59)
where

y τ = unfold(Y τ ) ∈ C I 1 I 3 ×1 , w τ = unfold(W τ ) ∈ C rI 3 ×1 , F Ωτ = P Ωτ F -1 I 3 ⊗ I I 1 ∈ C |Ωτ |×I 1 I 3
is the subsampled inverse Fourier transform, F n ∈ C n×n denotes the Discrete Fourier Transform matrix, the mixing matrix U ∈ R I 1 I 3 ×rI 3 is defined as U = F I 3 ⊗ I I 1 bcirc(U ) F -1 I 3 .

APPLICATIONS

Motivated by the so-called GROUSE algorithm for subspace tracking in [START_REF] Balzano | Online identification and tracking of subspaces from highly incomplete information[END_REF], TOUCAN applies the incremental gradient descent on the tensor Grassman manifold to track U t with time. It is worth noting that the objective function (5.59) is very common in subspace tracking problems.

Therefore, we can apply any subspace tracking algorithms which are capable of dealing with missing data to minimize (5.59) effectively.

Applications

Tensor tracking or dynamic tensor analysis has already been found several online applications and this section provides some typical examples in different research fields, from computer vision and neuroscience to anomaly detection.

Computer Vision

We begin this section with one of the earliest and most popular applications of tensor tracking:

visual tracking which is an important task in computer vision [START_REF] Smeulders | Visual tracking: An experimental survey[END_REF]. Naturally, video datasets can be represented as 4-th order streaming tensors of dimensionality, width × height × channel × time. Accordingly, there are several studies devoted to developing tensor-based visual trackers for better modeling the appearance of target objects, such as [START_REF] Li | Robust visual tracking based on incremental tensor subspace learning[END_REF][START_REF] Zhang | Visual tracking via dynamic tensor analysis with mean update[END_REF][START_REF] Hu | Semi-supervised tensor-based graph embedding learning and its application to visual discriminant tracking[END_REF][START_REF] Khan | An online spatio-temporal tensor learning model for visual tracking and its applications to facial expression recognition[END_REF], to name a few.

For example, Hu et al. in [START_REF] Li | Robust visual tracking based on incremental tensor subspace learning[END_REF] proposed the so-called IRTSA tracker using incremental tensor subspace learning to capture the appearance of objects. [START_REF] Li | Online robust low-rank tensor modeling for streaming data analysis[END_REF] introduced an online robust low-rank tensor modeling (ORLTM) method and found its success in video background subtraction. Andrews et al. in [START_REF] Sobral | Online stochastic tensor decomposition for background subtraction in multispectral video sequences[END_REF] developed an online stochastic tensor decomposition for background subtraction in multispectral video sequences. A robust streaming tensor-train algorithm was developed in [35] which also has the potential to detect foreground in video. Salut et al. in [START_REF] Salut | Online tensor robust principal component analysis[END_REF] proposed an online tensor robust principal component analysis and validated its effectiveness with the problem of background and foreground separation.

APPLICATIONS

In parallel, there are other interesting computer vision applications of dynamic tensor analysis, such as visual data recovery [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF][START_REF] He | Patch tracking-based streaming tensor ring completion for visual data recovery[END_REF], online video denoising [START_REF] Wen | Joint adaptive sparsity and low-rankness on the fly: an online tensor reconstruction scheme for video denoising[END_REF][START_REF] Wen | VIDOSAT: High-dimensional sparsifying transform learning for online video denoising[END_REF], and segmentation/classification [START_REF] Sun | Low-rank tucker approximation of a tensor from streaming data[END_REF][START_REF] Min | Inferring segmented dense motion layers using 5D tensor voting[END_REF].

Neuroscience

The brain can be viewed as a complex system with various interacting regions that can produce large multivariate data over time [START_REF] Bassett | Understanding complexity in the human brain[END_REF]. Many types of brain data can be represented by tensors, such as electroencephalography (EEG), magnetoencephalography (MEG), functional magnetic resonance imaging (fMRI), and near-infrared spectroscopy (NIRS) [START_REF] Cichocki | Noninvasive BCIs: Multiway signal-processing array decompositions[END_REF]. Apart from three intrinsic modes (i.e., frequency, channel, and time), brain data can have higher-order modes, such as, subjects, conditions, and trials [START_REF] Cichocki | Noninvasive BCIs: Multiway signal-processing array decompositions[END_REF]. Together with the fact that brain activities can change over time, dynamic tensor analysis has become an useful tool to study the structure and function of brain from such data.

In what follows, we list some appealing brain-computer interface applications to demonstrate the use of dynamic tensor analysis in neuroscience. First, for the problem of detecting dynamic functional connectivity networks (DFCNs), Ozdemir et al. in [START_REF] Ozdemir | Recursive tensor subspace tracking for dynamic brain network analysis[END_REF] introduced a recursive tensorbased framework capable of tracking DFCNs over time. The proposed framework was then applied for studying error-related negativity -a brain potential response when patients make errors during cognitive tasks [START_REF] Yeung | The neural basis of error detection: conflict monitoring and the error-related negativity[END_REF]. Mahyari et al. in [START_REF] Mahyari | A tensor decompositionbased approach for detecting dynamic network states from EEG[END_REF] developed a two-step approach using incremental tensor subspace analysis for detecting DFCNs. Particularly, they first detect change points at which the functional connectivity across subjects presents abrupt changes and then summarize DFCNs between successive change points. Recently, Acar et al. in [START_REF] Acar | Tracing evolving networks using tensor factorizations vs. ICA-based approaches[END_REF] proposed to use the Parafac2 model for tracking the evolution of connectivity networks and compared its performance with ICA and IVA. For the problem of localizing dynamic brain sources over time, Ardeshir et al. in [START_REF] Fotouhi | Evaluation of adaptive PARAFAC alogorithms for tracking of simulated moving brain sources[END_REF] utilized the boundary element method (BEM) [START_REF] Meijs | On the numerical accuracy of the boundary element method (EEG application)[END_REF] and the adaptive PARAFAC-RLST tracker [START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF] with two operational windowing schemes. A variant using augmented complex statistics in [START_REF] Fotouhi | Adaptive localization of moving eeg sources using augmented complex tensor factorization[END_REF] also has the ability to track moving EEG sources with time.

For the problem of online EEG completion, Trung et al. in [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF] proposed an adaptive CP algorithm called NL-PETRELS capable of tracking and imputing incomplete EEG data. Thanh et al. in [27,29] also demonstrated the use of ACP and RACP with real data by applying them for online EEG completion. Other neuroscience applications of tensor analysis were reviewed in [START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF][START_REF] Karahan | Tensor analysis and fusion of multimodal brain images[END_REF][START_REF] Zhou | Linked component analysis from matrices to high-order tensors: Applications to biomedical data[END_REF].

Anomaly Detection

Anomaly detection, which corresponds to identifying patterns and data points that do not conform to normal behavior, plays an essential role in many applications, such as cyber security, 5.8. CONCLUSIONS statistics, and finance, to name a few [START_REF] Chandola | Anomaly detection: A survey[END_REF]. Here, we provide some notable tensor-based anomaly detectors which are customized to specific online applications. of detecting multiple types of anomalies in road networks. We refer the readers to [START_REF] Fanaee-T | Tensor-based anomaly detection: An interdisciplinary survey[END_REF] for a broader interdisciplinary survey of tensors for anomaly detection.

Others

Apart from online applications in the domains above, tensor tracking also found success in some other research fields, namely wireless communications (e.g., channel tracking [START_REF] Yuan | Channel tracking for ris-enabled multi-user simo systems in time-varying wireless channels[END_REF], DOA tracking [START_REF] Luo | Sparse Bayes tensor and DOA tracking inspired channel estimation for V2X millimeter wave massive MIMO system[END_REF], and time delay estimation [START_REF] Garcez | Tensor-based subspace tracking for time-delay estimation in GNSS multi-antenna receivers[END_REF]), network analysis (e.g., link prediction [START_REF] Dunlavy | Temporal link prediction using matrix and tensor factorizations[END_REF],

internet scale monitoring [START_REF] Lin | SCENT: Scalable compressed monitoring of evolving multirelational social networks[END_REF], and bot activities and network intrusions [START_REF] Shin | Densealert: Incremental dense-subtensor detection in tensor streams[END_REF]), data analytics of chemical and biological manufacturing processes and components [START_REF] Sun | Opportunities in tensorial data analytics for chemical and biological manufacturing processes[END_REF][START_REF] Sanou | Online Nonnegative and Sparse Canonical Polyadic Decomposition of Fluorescence Tensors[END_REF], performance monitoring [START_REF] Meng | On-line monitoring of batch processes using a PARAFAC representation[END_REF][START_REF] Gourvenec | Monitoring batch processes with the STATIS approach[END_REF], and transportation [START_REF] Tan | Short-term traffic prediction based on dynamic tensor completion[END_REF][START_REF] Wang | Understanding urban dynamics via context-aware tensor factorization with neighboring regularization[END_REF].

Conclusions

Tensor tracking has recently gained increasing attention as a powerful tool for multidimensional data stream analysis. In this survey, we have provided a technical overview of online techniques for tracking streaming tensors over time. We highlighted the two most popular streaming CP and Tucker decompositions. Specifically, four main groups of streaming CP decomposition algorithms were emphasized, including subspace-based, block-coordinate descent, Bayesian inference, and multi-aspect streaming decompositions. We categorized the current streaming Tucker decomposition methods into three major classes based on their model architecture. They are online tensor dictionary learning, tensor subspace tracking, and multi-aspect streaming decompositions.

Recent years have also witnessed significant advances in other types of tensor decomposition such as tensor-train, BTD, and t-SVD. A brief survey on the existing methods which are capable of tracking tensors under these formats was presented.

INTRODUCTION

Tensor decomposition is a powerful multilinear algebra tool for analyzing multiway data and has been used

for various signal processing and machine learning applications. When the underlying tensor is derived from (multidimensional) data streams, streaming tensor decomposition or tensor tracking is required.

In

Introduction

The era of "Big Data", which deals with massive datasets, has brought new analysis techniques for discovering new valuable information hidden in the data [START_REF] Chen | Big data: A survey[END_REF]. Among these techniques is multilinear low-rank approximation (LRA) of matrices and tensors, which has recently attracted much attention from engineers and researchers [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF].

A tensor is a multidimensional array and provides a natural representation of multivariate and high-dimensional data. Low-rank approximation of tensors (t-LRA) can be considered as a multiway extension of LRA of matrices (which are two-way) to higher dimensions [START_REF] Kolda | Tensor decompositions and applications[END_REF]. Generally, t-LRA is referred to as tensor decomposition which allows factorizing a tensor into a sequence of basic components [START_REF] Kolda | Tensor decompositions and applications[END_REF]. As a result, t-LRA provides a useful tool for dealing with several largescale multidimensional problems in modern data analysis which would be, otherwise, intractable by classical methods. Two widely-used approaches for t-LRA are CANDECOMP/PARAFAC (CP) decomposition1 [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF] and Tucker decomposition [START_REF] Lathauwer | A multilinear singular value decomposition[END_REF]. Under CP decomposition, a tensor can be represented as a sum of rank-1 tensors; each rank-1 tensor is formulated as the outer product of vectors. Under Tucker decomposition, a tensor is factorized into a sequence of factor matrices acting on a reduced-size core tensor. "Workhorse" algorithms are based on the method 6.1. INTRODUCTION of alternating least-squares (ALS). The readers are referred to the work of [START_REF] Kolda | Tensor decompositions and applications[END_REF] for a good review.

Characteristics of big data are associated with the following three "V"s: high volume, high velocity and high veracity [START_REF] Chen | Big data: A survey[END_REF]. Velocity and veracity are the focus of this chapter. Velocity requires (near) real-time processing of data streams, while veracity demands robust algorithms to better deal with missing, noisy and inconsistent data. In online applications, data acquisition is often a time-varying process in which data are serially collected or changing with time. Besides, missing data are ubiquitous and more and more common in high-dimensional problems in which collecting all attributes of data is either too expensive or even impossible. In addition, outliers which are data points that appear to be inconsistent with or exhibit abnormal behaviour different from others causes cause several issues (e.g., they introduce bias in estimation) for knowledge discovery from data in general and data streams in particular. However, well-known t-LRA algorithms either face high complexity or operate in batch mode and, thus, may not be suitable for such problems. This has led to defining a variant of t-LRA, namely tensor tracking or streaming tensor decomposition.

Related Works

In the literature, there are several studies related to the problem of tracking online t-LRA in the missing data context; the tensors are said to be both streaming and incomplete. For adaptive CP decomposition, Mardani et al. proposed TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], which is a first-order algorithm and uses the method of stochastic gradient descent (SGD). Leveraging the framework of alternating minimization, TeCPSGD can estimate directly all factors but the one corresponding to the dimension growing over time in an efficient way. However, it often suffers from a slow convergence rate inherent to SGD and, hence, is not suitable for fast time-varying scenarios. To overcome this drawback, Kasai developed OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], which is an efficient second-order algorithm and exploits the recursive least-squares technique. OLSTEC provides a competitive performance in terms of estimation accuracy, but its computational complexity is much higher than that of TeCPSGD. In parallel, Chinh et al. proposed to first track the low-dimensional tensor subspace and then derive the loading factors from its Khatri-Rao structure [START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF]. However, the performance of this algorithm is sensitive to initialization. None of the abovementioned algorithms is capable of tracking online t-LRA when the tensors are of higher orders (i.e., greater than or equal to 4). On the other hand, some adaptive CP algorithms, such as [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF][START_REF] Smith | Streaming tensor factorization for infinite data sources[END_REF], are capable of handling higher-order streaming tensors. However, they do not handle incomplete datasets.

Recently, Zhang et al. have developed BRST [START_REF] Zhang | Variational bayesian inference for robust streaming tensor factorization and completion[END_REF], which is able to handle outliers. To track and separate the low-rank and sparsity components of the underlying tensor, a Bayesian statistical model was applied. The computational complexity of BRST is, however, very high and thus the method becomes inefficient when handling high-dimensional and fast-arriving data streams. the complex Grassmannian) [START_REF] Gilman | Online tensor completion and free submodule tracking with the t-SVD[END_REF], for tensor singular-value decomposition (t-SVD). Similar to RPTucker, TOUCAN also performs the incremental gradient descent on the Grassmann manifold.

However, both algorithms are only suitable for third-order tensors. Dimitris et al. have recently proposed the first robust online Tucker decomposition that can deal with streaming tensors in the presence of outliers [START_REF] Chachlakis | Dynamic L1-norm Tucker tensor decomposition[END_REF]. However, it was not designed for handling missing data. Some studies have been conducted to design efficient t-SVD algorithms for higher-order tensors, for example [START_REF] Martin | An order-p tensor factorization with applications in imaging[END_REF][START_REF] Zhang | Exact tensor completion using t-SVD[END_REF][START_REF] Jiang | Efficient multi-dimensional tensor sparse coding using t-linear combination[END_REF]. These algorithms were designed for batch computation and thus are not suitable for dynamic models. Recently, Thanh et al. have proposed TT-FOA [32], which is an adaptive tensor-train (TT) model for streaming tensors. Although TT-FOA and its stochastic version are capable of tracking the online low-rank tensor-train representation of large-scale and higher-order tensors, they were not designed to handle the situation with missing data.

In the multi-aspect streaming perspective of tensor analysis, Song et al. proposed an effective multi-aspect streaming tensor framework (MAST) [START_REF] Song | Multi-aspect streaming tensor completion[END_REF], used for dynamic tensor completion.

MAST can successfully track the multilinear LRA of incomplete tensors with dynamic growth in more than one tensor mode. A robust version of MAST for handling outliers, called outlierrobust multi-aspect streaming tensor completion and factorization (OR-MSTC), was proposed in [START_REF] Najafi | Outlier-robust multi-aspect streaming tensor completion and factorization[END_REF]. Thanks to the framework of alternating direction method of multipliers (ADMM), OR-MSTC can estimate the low-rank component from measurements corrupted by outliers. A new inductive framework, called SIITA, has been proposed to incorporate side information into incremental tensor analysis [START_REF] Madhav | Inductive framework for multiaspect streaming tensor completion with side information[END_REF]. SIITA can be seen as a counterpart of MAST for multi-aspect streaming Tucker decomposition. Although all these approaches provide good frameworks for the problem of dynamic tensor completion, they are either useful for third-order tensors only or are of high complexity and hence relatively inefficient in applications with online data streams.

In addition, convergence analysis of these algorithms is not available.

Some other studies attempted to extend robust subspace learning/online PCA for high-order tensor data.

Main Contributions

The main contributions of this chapter are summarized as follows:

• Firstly, under the CP format, we propose a novel adaptive CP (ACP) algorithm for tracking higher-order incomplete streaming tensors. ACP is fast and requires a low computational complexity and memory storage, thanks to the alternative minimization and randomized sketching. It can handle incomplete tensors derived from infinite data streams because it performs CP decomposition with constant time and space complexity that are independent of time index t. A convergence analysis is then provided to establish performance guarantees. To the best of our knowledge, the proposed ACP algorithm is the first one capable of dealing with streaming tensors of higher orders with "provable" convergence guarantee.

• Secondly, under the Tucker format, we propose the second algorithm, namely adaptive Tucker decomposition (ATD), more flexible than ACP, for the problem of online t-LRA.

ATD exhibits competitive performance in terms of both estimation accuracy and computational complexity. Its convergence guarantee is also presented. Also, this chapter presents for the first time a provable adaptive Tucker algorithm for this problem.

• Thirdly, we propose a novel method for robust adaptive CP, called RACP, for the robust tensor tracking problem in the presence of both missing data and outliers. Particularly, RACP aims to learn low-rank components of streaming tensors in an online fashion as well as offering robustness against gross data corruptions. RACP is a scalable and effective online CP algorithm with ability to (i) estimate low-rank components of streaming tensors derived from imperfect and noisy data streams due to missing observations and outlier corruptions, (ii) adapt the changes of the underlying data streams in dynamic and nonstationary environments, (iii) separate and reject sparse outliers in an online fashion with high accuracy, and (iv) easily incorporate prior information for dealing with specific constraints on the tensor model, e.g., smoothness and nonnegativity. Also, we prove that RACP is a provable adaptive CP algorithm with a convergence guarantee. Under mild conditions, we prove that the sequence of solutions generated by RACP converges asymptotically to a stationary point of the empirical loss function. Moreover, the asymptotic variation of the solutions and the almost-sure convergence of the objective function values are also analyzed.

• Last but not least, we provide several experiments on both synthetic and real data to illustrate the effectiveness of the proposed algorihtms in comparison with state-of-the-art tensor tracking algorithms.

6.2. TENSOR TRACKING WITH MISSING DATA 6.2 Tensor Tracking with Missing Data

Problem Statement

In this section, we investigate the problem of tracking an incomplete streaming tensor X t ∈ R I 1 ×I 2 ו••×I t N fixing all but the last dimension I t N (see illustration in Fig. 6.1 where the gray boxes represent missing data). Specifically, the t-th tensor slice

Y t ∈ R I 1 ×I 2 ו••×I N -1 of X t is
supposed to be generated under the following model:

Y t = P t ⊛ L t + N t , (6.1)
where P t is a binary observation mask, N t is a Gaussian noise tensor of the same size with Y t , and Y t is the multilinear low-rank component. The mask P t shows whether the (i 1 , i 2 , . . . , i N -1 )th entry of Y t is missing or not, i.e.,

p i 1 i 2 ...i N -1 =    1, if y i 1 i 2 ...i N -1 is observed, 0, otherwise. (6.2)
The low-rank component Y t is given by 2

L t ∆ = G; U (n) N -1 n=1 , u (N ) t , (6.3)
where r = [r 1 , r 2 , . . . , r N ] is the desired low multilinear rank, G ∈ R r 1 ×r 2 ו••×r N is the core tensor, ∈ R r N is the weight vector. 3 The underlying tensor X t is derived from appending the new slice X t to the previous X t-1 along the time dimension, i.e.,

U = {U (n) } N -1 n=1 with U (n) ∈ R
X t = X t-1 ⊞ N Y t , (6.4)
where I t N = I t-1 N + 1, as shown in Fig. 6.1.

The problem of tracking t-LRA of the incomplete streaming tensor X t can be stated as follows:

2 In online setting, the tensor core G and loading factors {U (n) } might be changing slowly over time, i.e.,

G = G t and U (n) = U (n) t , n = 1, 2, . . . , N -1.
Our algorithms are capable of estimating G and U accurately, but also successfully tracking their variation along the time. 3 In batch setting, the weight vector ut in (6.3) is seen as the t-th row of the last loading factor

U (N ) ∈ R I t N ×r N of X t.
6.2. TENSOR TRACKING WITH MISSING DATA Tensor Tracking with Missing Data: At each time t, we observe a streaming tensor slice Y t under the data model (6.1). We aim to estimate G t and U t that will provide a good multilinear low-rank approximation for X t in time.

t = 2 t = 1 K I J(t)
Applying batch methods to X t is possible, but these turns out inefficient for online (adaptive) settings. Our goal is to develop efficient one-pass algorithms, both in computational complexity and memory storage, for tracking the t-LRA of X t from past estimations at each time t.

In an adaptive scheme, we propose to minimize the following exponentially weighted cost function:

G t , U t = argmin G,U f t G, U = 1 t t τ =1 β t-τ ℓ G, U , P τ , Y τ , (6.5)
where the loss function ℓ(•) with respect to the τ -th temporal slice Y τ is given by

ℓ G, U , P τ , Y τ ∆ = min uτ ∈R r N P τ ⊛ Y τ -G, U (n) N -1 n=1 , u (N ) τ 2 F , (6.6)
and β ∈ (0, 1] is the forgetting parameter. Here, all observations (i.e. tensor slices) in the time interval [1, t] are taken into consideration in the estimation of the underlying low-rank component at each time t. The least-squares loss ℓ(.) defines the residual for each observation which measures the difference between the observed value and the estimated value of the tensor slice. β is used for discounting the effect of past observations exponentially, and ensuring that observations in the distant past are substantially down-weighted in the cost function relative to the latest ones.

Accordingly, when β < 1, this can facilitate the tracking ability of estimators, especially in timevarying and non-stationary environments. The effective window length for

β < 1 is (1 -β) -1
when t is large.

In the next two sections, we describe the two proposed algorithms for solving (6.5) under CP and Tucker decompositions. We make the following four assumptions for the convenience of deploying our algorithms as well as analyzing their performance.

(A1) Observed tensor slices {Y t } t≥1 are independent and identically distributed from a data-6.2. TENSOR TRACKING WITH MISSING DATA generating distribution, which is the underlying distribution of the dataset, having a compact set V. This assumption is very common for convergence analysis in online settings in general and adaptive tensor decomposition in particular, e.g., [25,[START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF][START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF]. Naturally, it holds in several scenarios, for instance, real-life data are often bounded such as image, video and audio. (A3) Observation mask tensors {P t } t≥1 are independent of {Y t } t≥1 and their entries obey the uniform distribution. With respect to the imputation of missing values and recovery of low-rank components, the uniform randomness allows the sequence of binary masks {P t } t≥1 to admit stable recovery [START_REF] Chatterjee | A deterministic theory of low rank matrix completion[END_REF]. Moreover, the number of observed entries in Y t is supposed to be larger than the lower bound O(rL log L), where L = I 1 I 2 . . . I N -1 and r = max(r 1 , r 2 , . . . , r N ) , and every row of Y (n) t is observed at least r entries for all n.

The constraints are fundamental conditions to prevent the underdetermined imputation problem [START_REF] Ashraphijuo | Fundamental conditions for low-CP-rank tensor completion[END_REF].

(A4) The low multilinear-rank model is either static or changing slowly over time, i.e., the core tensor and loading factors may vary slowly between two consecutive times t -1 and t:

G t ≃ G t-1 and U (n) t ≃ U (n) t-1 .
The tensor rank is supposed to be known.

Adaptive CP Decomposition

In this subsection, we first propose a fast adaptive CP algorithm for tracking online t-LRA of incomplete streaming tensors, called ACP. Then, we provide a performance analysis in terms of complexity and convergence to demonstrate its effectiveness and efficiency.

Proposed ACP Algorithm

Under the CP tensor model, (6.5) can be rewritten as follows:

U t = argmin U f t (U ) = 1 t t τ =1 β t-τ ℓ U , P τ , Y τ , (6.7)
where the loss function ℓ(U , P τ , Y τ ) is defined by

ℓ U , P τ , Y τ ∆ = min u (N ) τ ∈R r P τ ⊛ Y τ -U (n) } N -1 n=1 , u (N ) τ 2 F . (6.8)
Leveraging past estimations of the loading factors, we propose to minimize the surrogate g t (U ) of f t (U ) instead, which is defined, for a given value of u (N ) τ 1≤τ ≤t , by

g t (U ) = 1 t t τ =1 β t-τ P τ ⊛ Y τ -U (n) N -1 n=1 , u (N ) τ 2 F . (6.9)
The main motivation here stems from the following observations which will be detailed later in our convergence analysis. First, it is easy to verify that g t (U ) provides an upper bound on f t (U ) (i.e., f t (U ) ≤ g t (U ) for all U and a fixed set of {u

(N )
τ } 1≤τ ≤t ). Also, the error function e t (U ) = g t (U ) -f t (U ) is L-smooth for some constant L > 0, i.e. it is differentiable and ∇e t (U ) is L-Lipschitz continuous. As a result, g t (U ) is a first-order surrogate function of f t (U ) [START_REF]Incremental majorization-minimization optimization with application to largescale machine learning[END_REF] and hence its theoretical convergence results can be achieved without making any strong assumptions on f t (U ). In particular, the sequence of surrogate values {g t (U t )} ∞ t=1 is quasi-martingale and converges almost surely. Accordingly, under a simple assumption that the directional derivative of f t exists in any direction at any U , {g t (U t )} ∞ t=1 and {g t (U t )} ∞ t=1 converge to the same limit. Indeed, the solution U t derived from minimizing g t (U ) converges to a stationary point of f t (U ) when t approaches infinity. Furthermore, g t (U ) can be effectively minimized with a convergence rate of O(1/t) and it is much simpler than minimizing f t (U ).

In order to obtain a low-complexity estimator, we exploit that (6.9) can be efficiently solved using the alternating minimization framework whose iteration step coincides with the tensor slice's acquisition in time. In particular, it can be divided into two main stages: (i) estimate u (N ) t first, given the old estimation U t , and (ii) update the loading factor U (n) t , given u (N ) t and the remaining factors. The proposed ACP algorithm is summarized in Algorithm 6. In the following, we will describe the key steps of our algorithm for minimizing (6.9).

Step 1: Estimation of u (N ) t

Under the assumption that the loading factors might be static or slowly time-varying, i.e., U t ≃ U t-1 , the weight vector u (N ) t can be derived from the loss function ℓ(.) in (6.8) at time t by

u (N ) t = argmin u∈R r P t ⊛ Y t -H t × N u ⊤ 2 2 , ( 6
H t = N -1 n=1 U (n) t-1 . (6.12)
For the sake of convenience, let Ω t and x Ωt be the set and vector containing the observed entries of Y t , while H Ωt is the sub-matrix of H t obtained by selecting the rows corresponding to

x Ωt .

Generally, problem (6.11) is an overdetermined least-squares (LS) regression and requires O(|Ω t |r 2 ) with respect to (w.r.t.) computational complexity to compute the exact LS solution [START_REF] Raskutti | A statistical perspective on randomized sketching for ordinary least-squares[END_REF]. Thus, it costs time and effort when dealing with high-dimensional and high-order tensors.

We propose to solve a regularized least-squares sketch of (6.11) instead, i.e., u

(N ) t = argmin u∈R r L y Ωt -H Ωt u 2 2 + α u 2 2 , (6.13)
where α is a small positive parameter for regularization, L(.) is a sketching map that helps reduce the sample size, and hence speed up the calculations. Here, the introduction of α∥u∥ 2 2 is for avoiding the singular/ill-posed computation or pathological cases as well as increasing the least-squares interpretability in practice. 5 Accordingly, the updated rule for u t is given by

u (N ) t = H ⊤ St H St + αI -1 H ⊤ St x St , (6.14)
where H St and x St are transformed versions of H Ωt and x Ωt under the sketching L(.), respectively.

In what follows, we indicate that in many cases, the uniform row-sampling can provide a good sketch for (6.11) in which each row has equal chance of being selected. We start by revisiting the definition of the leverage scores and coherence of a matrix. . . . ; a ⊤ m ] ∈ R m×r with m > r, its i-th row leverage score is defined as

T i (A) ∆ = a ⊤ i A ⊤ A # a i = U A (i, :) 2 2 , i = 1, 2, . . . , m. (6.15)
Here, U A ∈ R m×r is the left singular vector matrix of A. The coherence of A is the largest leverage score

µ(A) = max i T i (A). (6.16)
The leverage score T i (A) evaluates the contribution of a i in constituting A's row space.

Accordingly, if the value of µ(A) is high, A contains at least one "strong" row whose removal would have a pernicious effect on its row space. When the value of µ(A) is small (e.g. µ(A) ≈ r/m ≪ 1), no specific row is more important than others, i.e. information is approximately uniformized across all rows. In such a case, the matrix A is called incoherent. The following proposition indicates that the Khatri-Rao structure of H t may increase the incoherence from its factors.

Proposition 11

(Coherence of H t ). Let μt-1 = 1 N -1 N -1 n=1 µ U (n) t-1 . We have µ(H t ) = µ N -1 n=1 U (n) t-1 (i) ≤ N -1 n=1 µ U (n) t-1 (ii) ≤ μN-1 t-1 < 1. (6.17)
Proof. The first inequality (i) is indeed a corollary of Lemma 4 in [START_REF] Battaglino | A practical randomized CP tensor decomposition[END_REF] which shows that µ(A 1 ⊙

A 2 ) ≤ µ(A 1 )µ(A 2 ) for any A 1 and A 2 of suitable sizes.

The second inequality (ii) is obtained by applying the AM-GM inequality to the set of N

positive numbers µ U (n) t-1 M -1 n=1 .
Accordingly, when dealing with a high-order streaming tensor (N is large) and/or with some incoherent tensor factors, µ(H t ) ≤ μN-1 t-1 ≪ μt-1 < 1, i.e., H t has low coherence. In such cases, uniform row-sampling is effective [START_REF] Avron | Blendenpik: Supercharging LAPACK's leastsquares solver[END_REF][START_REF] Ipsen | The effect of coherence on sampling from matrices with orthonormal columns, and preconditioned least squares problems[END_REF]. In the presence of highly coherent factors, a preconditioning (mixing) step is necessary to guarantee the incoherence. For instance, the subsampled randomized Hadamard transform (SRHT) is a good candidate which can produce a transformed matrix whose rows have (almost) uniform leverage scores [START_REF] Tropp | Improved analysis of the subsampled randomized Hadamard transform[END_REF]. In this context, we here emphasize that well-known randomized LS algorithms can help save much computational complexity while obtaining reasonable estimations of u (N ) t , especially for large-scale low-rank tensors.

TENSOR TRACKING WITH MISSING DATA

Step 2: Estimation of U (n) t

The loading factor U (n) t can be updated by minimizing the objective function g t (•) w.r.t. U (n) , as

U (n) t = argmin U (n) ∈R In×r 1 t t τ =1 β t-τ P (n) τ ⊛ Y (n) τ -U (n) W (n) τ ⊤ 2 F , (6.18)
where

Y (n) τ (resp. P (n)
τ ) is the mode-n unfolding of Y τ (resp. P τ ) and the coefficient matrix

W (n) τ
is given by

W (n) τ = N -1 i=1,i̸ =n U (i) t-1 ⊙ (u (N ) τ ) ⊤ . (6.19)
Interestingly, we exploit the fact that minimization (6.18) can boil down to the problem of subspace tracking in the presence of missing data [START_REF] Balzano | Streaming PCA and Subspace Tracking: The Missing Data Case[END_REF]. Particularly, the solution of (6.18) can be obtained by minimizing subproblems for each row u

(n) m of U (n) , m = 1, 2, . . . , I n as u (n) t,m = argmin u (n) m ∈R r 1 t t τ =1 β t-τ P (n) τ,m y (n) τ,m ⊤ -W (n) τ u (n) m ⊤ 2 F , (6.20)
where

y (n) τ,m is the m-th row of Y (n) τ
and the row-mask matrix

P (n) τ,m = diag(P (n)
τ (m, :)). Thanks to the parallel scheme of the well-known PETRELS algorithm for subspace tracking [START_REF] Chi | PETRELS: Parallel subspace estimation and tracking by recursive least squares from partial observations[END_REF], we derive an efficient estimator for minimizing the exponentially weighted LS cost function (6.18).

Particularly, we first define two auxiliary matrices

S (n) t and V (n) t as follows 6 S (n) t = βS (n) t-1 + W (n) t ⊤ W (n) t , (6.21)
V (n) t = S (n) t -1 W (n) t ⊤ . (6.22)
The loading factor U (n) t is then updated recursively by

U (n) t = U (n) t-1 + ∆Y (n) t V (n) t ⊤ , (6.23)
where the matrix ∆Y

(n) t
is derived from the mode-n unfolding of the residual error tensor ∆Y t

∆Y t = P t ⊛ Y t -H t × N (u (N ) t ) ⊤ . (6.24)
This is not PETRELS, but a modified version. Here, we can utilize the already updated U

(n) t for tracking the remaining factors which can improve the rate of convergence. Also, we can estimate 6 To enable the recursive updating rule, the matrix S (n) 0 is initialized by a scaled identity matrix S

(n) 0

= δnIr n with δn > 0.

TENSOR TRACKING WITH MISSING DATA

Algorithm 6: Adaptive CP Decomposition (ACP)

Input: Incomplete slices P t ⊛ Y t ∞ t=1 , Y t ∈ R I1×I2ו••×I N -1 ×1
, CP rank r, Forgetting factor β ∈ (0, 1], Parameters: α > 0, δ > 0, and m > 0.

Initialization: U (n) 0 N -1 n=1 is initialized randomly and S (n) 0 N -1 n=1 = δI r . Main Program: Procedure: for t = 1, 2, . . . do Y Ωt = P t ⊛ Y t
Step 1: Estimation of u t S = randsample |Ω t |, ⌊mr log r⌉

H t = I N -1 n=1 × n U (n) t-1 u t = H ⊤ St H St + αI -1 H ⊤ St y St U (N ) t = U (N ) t-1 ⊤ , u (N ) t ⊤ ∆Y t = P t ⊛ Y t -H t × N u ⊤ t Step 2: Estimation of U (n) t N -1 n=1 for n = 1, 2, . . . , N -1 do Y (n) Ωt = unfold n (Y Ωt ) ∆Y (n) t = unfold n (∆X t ) W (n) t = U (n) t-1 # Y (n) Ωt ⊤ S (n) t = βS (n) t-1 + W (n) t ⊤ W (n) t V (n) t = (S (n) t ) -1 W (n) t U (n) t = U (n) t-1 + ∆Y (n) t V (n) t ⊤ end end Output: U (n) t N n=1
all the N factors in a parallel scheme which reduces further the cost when several computational units are available.

Performance Analysis Memory Storage and Computational Complexity

For the sake of simplifying the analysis, we assume that the fixed dimensions of the streaming tensor Y t are equal to I and the CP rank is much lower than I, r ≪ I.

TENSOR TRACKING WITH MISSING DATA

With respect to memory storage, ACP requires O (N -1)(Ir + r 2 ) words of memory at each time t, in particular for N loading factors {U (n) } N -1 n=1 and N -1 matrices S

(n) t of size r × r.

In terms of computational complexity, the estimation of u is of size r × r, thus the computation of S (n) t -1 is not expensive and it is independent of the tensor dimension. In conclusion, the overall computational complexity is

O |Ω t |r + ((N -1)I N -2 + |S t |)r

Convergence Guarantee

Inspired by our companion work on robust subspace tracking in [25] and the convergence analysis for 3-order tensors in [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], we derive a unified approach to analyze the convergence behavior of ACP for high-order streaming tensors with missing data. Specifically, we analyze the convergence of both the sequence of objective values {f t (U t )} ∞ t=1 and the sequence of generated solutions {U t } ∞ t=1 . Our main theoretical result is stated in the following lemma.

Lemma 9. Given assumptions (A1)-(A4), β = 1, and the true U is fixed, the sequence of solutions {U t } ∞ t=1 generated by ACP converges to a minimum point of f t when t → ∞.

Proof Sketch. Our proof contains three main stages: (S1) we show that the solutions {U t , u t } ∞ t=1 are uniformly bounded to justify the well-definedness condition. Their variations between two successive time instances satisfy ∥U

(n) t+1 -U (n) t ∥ F → O(1/

Adaptive Tucker Decomposition

The proposed ACP algorithm is not always well-defined due to the fact that for a given CP rank, the optimal CP-based representation of tensors may be nonexistent [START_REF] Silva | Tensor rank and the ill-posedness of the best low-rank approximation problem[END_REF]. Under the Tucker format, we now propose a more flexible algorithm called adaptive Tucker decomposition (ATD).

In the same way, we propose to minimize the following surrogate function g t (G, U ) of f t (G, U ) 6.2. TENSOR TRACKING WITH MISSING DATA Algorithm 7: Adaptive Tucker Decomposition (ATD)

Input: Observations P t ⊛ X t ∞ t=1 , X t ∈ R I1×I2ו••×I N -1 ×1 , Tucker rank r TD = [r 1 ,
Y Ωt = P t ⊛ Y t Step 1: Estimation of u t S = randsample |Ω t |, ⌊mr N log r N ⌉ H t = G t-1 N -1 n=1 × n U (n) t-1 u t = H ⊤ St H St + αI -1 H ⊤ St y St U (N ) t = U (N ) t-1 ⊤ , u (N ) t ⊤ ∆Y t = P t ⊛ Y t -H t × N u ⊤ t Step 2: Estimation of {U (n) t } N -1 n=1 for n = 1, 2, . . . , N -1 do W (n) t = U (n) t-1 # Y (n) Ωt S (n) t = βS (n) t-1 + W (n) t W (n) t ⊤ V (n) t = S (n) t -1 W (n) t U (n) t = U (n) t-1 + ∆Y (n) t V (n) t ⊤ end Step 3: Estimation of G t Z t = u t ⊗ N -2 n=2 U (n) t ∆G t = U (1) t # ∆Y (1) t Z # t ∆G t = reshape(∆G t , r TD ) G t = G t-1 + ∆G t end Output: U (n) t N n=1 and G t in (6.5): {G t , U t } = argmin G,U g t (G, U ) = 1 t t τ =1 β t-τ P τ ⊛ Y τ -G; U (n) N -1 n=1 , u (N ) τ 2 F , (6.25)
to leverage old estimations of the tensor core and the loading factors at each time t.

TENSOR TRACKING WITH MISSING DATA

Proposed ATD Algorithm

Thanks to the alternating minimization framework, we can obtain an efficient first-order estimator for optimizing (6.25) in the same manner as ACP. Specifically, we first update the weight vector u

(N )
t , given old estimations of G and U , then estimate the loading factors {U

(n) t } n≥1 given u (N )
t , G t-1 and the remaining factors, and finally obtain the core tensor G t from the latest updated factors. The proposed algorithm is summarized in Algorithm 7.

Step 1: Estimation of u

(N ) t
We can derive the weight vector u (N ) t from the minimizing the last summand of g t (G, U ) as follows:

u (N ) t = argmin u∈R r N P t ⊛ Y t -H t × N u ⊤ 2 2 , (6.26)
where

H t = G t-1 N -1 n=1 × n U (n)
t-1 . Similar to (6.10), the expression (6.26) can be readily reformulated into its matrix-vector format as follows:

u (N ) t = argmin u∈R r N P t y t -H t u 2 2 , (6.27)
where y t = vec(Y t ), H t is the unfolding matrix of the tensor H t and the observation matrix P t = diag(vec(P t )). The closed-form solution of (6.27) can be directly obtained by applying the LS method as

u (N ) t = H ⊤ t P t H t + αI -1 H ⊤ t P t y t , (6.28)
where α > 0 is a small regularization parameter to avoid pathological cases in practice.

In order to speed up the computation of (6.28), the same randomized sampling technique as in (7.12) can be applied to obtain an approximated version of u

(N ) t .
Step 2: Estimation of

U (n) t
Given u t and old estimations of G t-1 and U t-1 , we rewrite the minimization (6.25) with respect to the variable U (n) as follows: τ is the mode-n unfolding of the tensor W τ which is defined by

U (n) t = argmin U (n) ∈R In×rn 1 t t τ =1 β t-τ P (n) τ ⊛ Y (n) τ -U (n) W (n) τ 2 F , ( 6
W τ = G t-1 N -1 i=1,i̸ =n × i U (i) t-1 × N +1 (u (N ) τ ) ⊤ . (6.30)
Minimization (6.29) is similar to its counterpart in the proposed ACP algorithm in (6.18). Therefore, we can apply the same subspace-based technique to update U (n)

t . In particular, the updating rule for U (n) t can be given by

U (n) t = U (n) t-1 + ∆Y (n) t (V (n) t ) ⊤ , (6.31)
where the residual error ∆Y

(n) t

and the coefficient matrix

V (n) t
are computed as

∆Y (n) t = P (n) t ⊛ Y (n) t -U (n) t-1 W (n) t , (6.32) V (n) t = S (n) t -1 W (n) t , (6.33)
where the matrix S

(n) t is updated recursively as

S (n) t = βS (n) t-1 + W (n) t (W (n) t ) ⊤ . (6.34)
Step 3: Estimation of G t

For the estimation of G t given the latest updated loading factors, (6.25) is reformulated as

G t = argmin G 1 t t τ =1 β t-τ P (1) τ ⊛ Y (1) τ -U (1) t G (1) Z τ 2 F , (6.35)
where the variable G (1) is the mode-1 unfolding of G and the matrix Z τ is given by

Z τ = u (N ) τ ⊗ N -1 n=2 U (n) t . (6.36)
When handling a streaming tensor with a huge number of slices (i.e., t is large) and a large number of unknown parameters in G (i.e., N n=1 r n is large), applying batch gradient methods for (6.35) may be time-consuming despite the effect of the forgetting factor λ. Stochastic approximation is introduced as a good alternative [START_REF] Spall | Introduction to Stochastic Search and Optimization[END_REF].

In particular, we minimize the last summand of (6.35) instead:

G t = argmin G P (1) t ⊛ Y (1) t -U (1) t G (1) Z t 2 F .
(6.37)

TENSOR TRACKING WITH MISSING DATA

Given the estimation of U t , the residual error between the newcoming tensor slice and the recovered one is given by ∆Y (1)

t = P (1) t ⊛ Y (1) t -U (1)
t G

(1)

t-1 Z t . (6.38)
Accordingly, we can derive the variation of G at time t from

∆Y

(1)

t = P (1) t ⊛ U (1)
t ∆G

(1)

t Z t , (6.39)
where ∆G

(1)

t = G (1) t -G (1)
t-1 . In particular, ∆G t is computed as 7

∆G

(1)

t = U (1) t # ∆Y (1) t Z # t . (6.41)
After that, ∆G

t will be reshaped into a tensor ∆G t of size

r 1 × r 2 × • • • × r N .
To sum up, we obtain the simple rule for updating G t as follows:

G t = G t-1 + ∆G t . (6.42)
We note that for overdetermined cases, the rule for updating G t can be sped up by using the following "vector trick" in [START_REF] Van Loan | The ubiquitous Kronecker product[END_REF]:

vec(ABC ⊤ ) = (C ⊗ A)vec(B). (6.43)
In particular, the expression (6.39) can be cast into the standard least-squares format as follows:

δy t = P t u t ⊗ N -1 n=1 U (n) t δg t , (6.44)
where δx t = vec(∆Y

t ), δg t = vec ∆G

(1) t and P t = diag vec(P

t ) . Interestingly, (6.44) is of the Kronecker structure, thus δg t can be efficiently computed by applying randomized sketching techniques with a much lower complexity, e.g., the uniform sampling or the Kronecker product regression in [START_REF] Diao | Optimal sketching for Kronecker product regression and low rank approximation[END_REF]. 7 Since Zt is of the Kronecker structure, we can obtain the pseudoinverse of Zt efficiently by using the following nice property [START_REF] Langville | The Kronecker product and stochastic automata networks[END_REF]

(A1 ⊗ A2 ⊗ • • • ⊗ An) # = A # 1 ⊗ A # 2 ⊗ • • • ⊗ A # n .
(6.40)

TENSOR TRACKING WITH MISSING DATA

Step 4: Orthogonalization Step (Optional)

In the cases where the orthogonality constraints are imposed on the loading factors, we add an orthogonalization step of U (n) at each time t as follows:

U (n) t = U (n) t U (n) t ⊤ U (n) t -1/2 , ( 6.45)
where (.) -1/2 represents the inverse square root or simply take the QR decomposition of U t .

Accordingly, the update of ∆G t in (6.41) can be speeded up by replacing the pseudo-inverse with the transpose operator:

∆G t = U (1) t ⊤ ∆Y (1)
t Z ⊤ t . (6.46)

Performance Analysis Memory Storage and Computational Complexity

We assume that the fixed dimensions of the streaming tensor are equal to I and the desired Tucker rank is r TD = [r, r, . . . , r].

In terms of memory storage, ATD requires O(r N ) and O (N -1)Ir words of memory for saving the core tensor G and N -1 loading factors {U (n) } N -1 n=1 respectively. In addition, the cost for saving N -1 matrices S (n) t is O((N -1)r 2 ) words of memory in total.

In terms of computational complexity, the computation of ATD comes from three main estimations: (i) the weight vector u flops where S 2 is the set of selected samples from (6.44). Therefore, the overall computational

complexity of ATD is O |Ω t |r + (I N -2 + |S 1 |)r 2 + |S 2 |r 2N in parallel scheme.

Convergence Guarantee

The convergence of ATD can be stated by the following lemma: Lemma 10. Given assumptions (A1)-(A4), β = 1, the true G and U are fixed, the solutions {G t , U t } ∞ t=1 generated by ATD converges to a stationary point of the empirical cost function f t when t → ∞.

The proof of Lemma 10 can be obtained by applying the same arguments and principles as in the case of ACP, detailed in the Appendix A. In particular, the analysis consists of the following three main stages: (S1) the surrogate function g t (G, U ) is strongly bi-convex in the sense that G and U are seen as multivariate variables. Solutions {G t , U t } ∞ t=1 generated by ATD are bounded and their variations between two successive time instances satisfy ∥U

(n) t+1 -U (n) t ∥ F → O(1/t) a.s. (S2) The nonnegative sequence {g t (G t , U t )} ∞
t=1 is quasi-martingale and hence convergent almost surely. Furthermore,

g t (G t , U t ) -f t (G t , U t ) → 0 a.s. (S3) The empirical cost function f t (G, U )
is continuously differentiable and Lipschitz. The sequence of solutions {G t , U t } ∞ t=1 converges to a stationary point of f t (G, U ), i.e., when t → ∞, the gradient ∇f t (G t , U t ) → 0 a.s.

Tensor Tracking with Sparse Outliers

Problem Statement

Here, we consider an incomplete streaming tensor X t ∈ R I 1 ×I 2 ו••×I N -1 ×t whose slices are serially observed with time. At each time t, X t is particularly obtained by concatenating a new incoming

"slice" Y t ∈ R I 1 ×I 2 ו••×I N -1 ×1
into the previous X t-1 along the time dimension, i.e., X t = X t-1 ⊞ N Y t . Particularly, we suppose to observe the slice Y t satisfying the following model:

Y t = P t ⊛ L t + O t + N t , (6.47)
where P t is a binary mask tensor, L t is a low-rank tensor, O t is a sparse tensor containing outliers, N t is a Gaussian noise tensor, and all these tensors are of the same size with Y t , please see Fig 6 .2 for an illustration.

Specifically, the observation mask P t indicates whether the (i 1 , i 2 , . . . , i N -1 )-th entry of L t is observed or missing, i.e.,

p i 1 i 2 ...i N =    0, if x i 1 i 2 ...i N -1 is missing, 1, otherwise.
(6.48)

TENSOR TRACKING WITH SPARSE OUTLIERS

The low-rank tensor L t is generated according to the following model:

L t = U (n) N -1 n=1 , u (N ) t , (6.49)
where u

(N ) t ∈ R r×1 is a weight vector 8 and

{U (n) } N -1 n=1 , with U (n) ∈ U n ⊆ R In×r , are loading factors. For short, write D := U 1 × U 2 × • • • × U N and denote D = (U (1)
) ⊤ , . . . , (U (N ) ) ⊤ ⊤ the tensor dictionary containing all loading factors. The robust tensor tracking (RTT) problem can be stated as follows:

Tensor Tracking with Sparse Outliers: At each time t, we observe a streaming tensor slice Y t under the data model (6.47). We aim to estimate D t ∈ D such that it can provide a good multilinear low-rank approximation for X t in time. Now, we define a loss function ℓ(•) that not only promotes sparsity but also preserves convexity.

For a fixed D and a tensor slice X under a binary observation mask P, the loss function w.r.t. D and {P, Y} is defined as

ℓ D, P, X = min u,O l D, P, X , O, u , with (6.50) l D, P, Y, O, u = O 1 + ρ 2 P ⊛ Y -O -H × N u ⊤ 2 F , (6.51)
where n) . The ℓ 1 -norm is to promote the sparsity on O and ρ > 0 is a regularized parameter. Now, given a streaming set of incomplete tensor slices {P τ ⊛ Y τ } t τ =1 , robust tensor tracking (RTT) can be formulated as the following optimization problem:

H = I N -1 n=1 × n U (
D t = argmin D f t (D) = 1 L t t τ =t-Lt+1 β t-τ ℓ D, P τ , Y τ , (6.52)
where L t is the length of a sliding window and β is a forgetting factor. When L t = t, β = 1, the minimization of (6.52) boils down to its counterpart in batch setting. When 0 < L t < t or β < 1, it reduces the impact of past observations, and hence facilitates the tracking ability of RTT estimators in time-varying conditions.

We make some assumptions to support the proposed algorithm in Section III. First, entries of tensor slices {Y t } t≥1 are Frobenius-norm bounded, i.e., ∥Y t ∥ F ≤ M x < ∞ ∀t. This prevents arbitrarily large values in observations and ill-conditioned computation. Next, the tensor rank r is supposed to remain unchanged over time. In addition, tensor factors {U (n) t } N n=1 are bounded and full column rank, i.e., rank(U

(n) t ) = r < I n and ∥U (n) t ∥ F ≤ κ U < ∞ ∀n.
Besides, the 6.3. TENSOR TRACKING WITH SPARSE OUTLIERS variation between two consecutive time instants is small, U

(n) t ≃ U (n) t-1 ∀n, t i.e. D t-1 ≃ D t .
This assumption permits the estimation of the outliers and the coefficient vector from the previous estimation with reasonable accuracy. Under these assumptions, our optimization algorithm is capable of accurately estimating tensor factors, but also successfully tracking their variation along the time.

Robust Adaptive CP Decomposition

In this section, we first propose the robust adaptive CP (RACP) algorithm for the RTT problem in the presence of missing data and outliers. Then, we introduce two simple extensions of RACP in order to deal with smoothness condition and nonnegative constraints.

Proposed RACP Algorithm

Solving the minimization of (6.52) exactly is possible but difficult since f t (•) is nonconvex. We here adapt it using the majorization-minimization (MM) framework [START_REF]Incremental majorization-minimization optimization with application to largescale machine learning[END_REF], which has been successfully applied to several signal processing problems in general [START_REF] Sun | Majorization-minimization algorithms in signal processing, communications, and machine learning[END_REF] and online learning problems in particular [25,[START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF][START_REF] Feng | Online robust PCA via stochastic optimization[END_REF][START_REF] Guan | Online nonnegative matrix factorization with robust stochastic approximation[END_REF]. In essence, we decompose it into two main stages: (i) online outlier rejection and (ii) tensor factor tracking.

On the arrival of Y t at each time t, we first estimate the outlier tensor O t and the coefficient vector u t based on the old estimation D t-1 . Specifically, we solve the following optimization:

{O t , u t } = argmin O,u l D t-1 , P t , Y t , O, u . (6.53)
From the past statistics {D τ , P τ , Y τ , O τ , u τ } τ ≥1 , the set of loading factors

D t = {U (n)
t } N n=1 can be updated by minimizing the following majorizing surrogate ft (•):

ft (D) = 1 L t t τ =t-Lt+1 β t-τ l D, P τ , Y τ , O τ , u τ , ( 6
Input: Tensor slices Pt ⊛ Y t ∞ t=1 , Y t ∈ R I 1 ×I 2 ו••×I N -1 ×1
, rank r, forgetting factor β ∈ (0, 1], Parameters: penalty ρ > 0, precision ϵ res , ϵ out > 0, maximum iteration K, α ∈ [1.5, 1.8], δ > 0.

Initialization: U (n) 0 N -1 n=1 is initialized randomly and S (n) 0 N -1 n=1 = δIr. Main Program: Procedure: for t = 1, 2, . . . do Stage 1: Online Outlier Rejection Ht-1 = N -1 n=1 U (n) t-1 o 0 , z 0 , u 0 ← 0 for i = 1, 2, . . . , K do u i = H ⊤ t-1 PtHt-1 # H ⊤ t-1 Pt yt -o i-1 -z i-1 /ρ , r i = αPt yt -Ht-1u i + (1 -α)o i-1 o i = S 1/ρ r i -z i-1 /ρ , z i = z i-1 + ρ(o i -r i ), if stopping criteria are met break end Outlier Removal (Re-update of Pt in (6.61) is optional): Y t = Pt ⊛ (Y t -Ot) Stage 2: Estimation of U (n) t N n=1 for n = 1, 2, . . . , N do W (n) t = N i=1,i̸ =n U (i) t-1 ⊙ (u (N ) t ) ⊤ [Jacobi] W (n) t = n-1 i=1 U (i) t ⊙ N i=n+1 U (i) t-1 ⊙ (u (N ) t ) ⊤ [Gauss-Seidel] W (n) t = W (n) t ⊤ W (n) t-L t ⊤ ⊤
for m = 1, 2 . . . , In do

P (n) t,m = P (n) t,m 0 0 -β L t P (n) t-L t ,m Ỹ(n) t,m = Ŷ(n) t,m Ŷ(n) t-L t ,m S (n) t,m = βS (n) t-1,m + W (n) t ⊤ P (n) t,m W (n) t V (n) t,m = S (n) t,m -1 W (n) t ⊤ δ ỹ(n) t,m = P (n) t,m ỹ(n) t,m ⊤ -W (n) t u (n) t-1,m ⊤ u (n) t,m = u (n) t-1,m + δ ỹ(n) t,m ⊤ V (n) t,
U (n) t :,r = U (n) t :,r U (n) t :,r 2 2 . Re-estimation of ut: ut = H ⊤ t PtHt # H ⊤ t Pt(xt -ot) where Ht = N n=1 U (n) t end Output: Loading factors U (n) t N n=1 .

Stage 1: Online Outlier Rejection

To estimate O t and u (N ) t , we recast (6.53) into the following standard matrix-vector form:

o t , u (
= N -1 n=1 U (n) t-1 .
Since both terms of (6.55) are convex, it can be efficiently solved by several methods with convergence guarantees. Here, we use an ADMM solver to minimize (6.55) due to its simple interpretation and moderate convergence rate [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. At the i-th iteration, we particularly read

u i = H ⊤ t-1 P t H t-1 # H ⊤ t-1 P t y t -o i-1 -z i-1 /ρ , (6.56
)

r i = αP t y t -H t-1 u i + (1 -α)o i-1 (6.57
)

o i = S 1/ρ r i -z i-1 /ρ , (6.58)
z i = z i-1 + ρ(o i -r i ), (6.59)
where S(•) is the soft-thresholding operator of the ℓ 1 -norm defined as S α (x) = max(0, x -α)max(0, -x -α) and α ∈ [1.5, 1.8] is a relaxation parameter. The procedure is stopped when residuals are small, i.e., ∥P t (y t -H t-1 u io i )∥ 2 ≤ ϵ res and ∥o ir i ∥ 2 ≤ ϵ out where ϵ res , ϵ out > 0 are predefined accuracy parameters or when the procedure reaches the maximum number of iterations.

After the sparse outlier O t is detected, we reduce the effect of O t on the tracking process by the following outlier removal

Y t = P t ⊛ (Y t -O t ). (6.60)
In some cases, we can skip the corrupted entries in Y t by re-updating the mask P t as

p i 1 i 2 ...i N =    0, if x i 1 ...i N is missing or outlier, 1, otherwise. (6.61)
Here, the removal step (6.60) still holds under the new binary mask P t . This approach stems from the following observations. In the context of subspace tracking (ST), rejecting outliers can facilitate the tracking ability of ST estimators since only "clean" measurements involve the process [25]. Our next stage for estimating the tensor basis can indeed boil down to the ST problem with missing data, so the outlier rejection mechanism of (6.61) can improve performance.

Please see Fig. 6.24 for an illustration that the outlier rejection mechanism can help improve the convergence rate of RACP when the fraction of corrupted entries is not too large.

Stage 2: Estimation of factors

U (n) t N -1 n=1
The optimization (6.54) can be effectively solved by using the block-coordinate descent (BCD)

technique. The main idea is to minimize alternately the surrogate ft (

U (n) t = argmin U (n) t τ =t-Lt+1 β t-τ P (n) τ ⊛ Y (n) τ -U (n) W (n) τ ⊤ 2 F , (6.63)
where Y

(n) τ and P

(n) τ are the mode-n unfoldings of Y τ and P τ , and

W (n) τ
is given by

W (n) τ =              N -1 i=1,i̸ =n U (i) t-1 ⊙ (u (N ) τ ) ⊤ [Jacobi], n-1 i=1 U (i) t ⊙ N -1 i=n+1 U (i) t-1 ⊙ (u (N ) τ ) ⊤ [Gauss-Seidel]. (6.64)
Depending on the implementation, we can use one of the two iterative methods: the Jacobi scheme supports the parallel and/or distributed processing while the Gauss-Seidel scheme is useful for a sequential (serial) one. Excepting the closed-form of W

τ , both methods share the same procedure for solving (6.63) which is detailed as follows.

The minimization of (6.63) can be decomposed into sub-problems for each row u

(n) m of U (n) , m = 1, 2, . . . , I n , as u (n) t,m = argmin u (n) m t τ =t-Lt+1 β t-τ P (n) τ,m ŷ(n) τ,m ⊤ -W (n) τ u (n) m ⊤ 2 F , (6.65)
where

ŷ(n) τ,m is the m-th row of Y (n)
τ , and the row-mask matrix is given by P

(n) τ,m = diag P (n) τ (m, :) .
The optimal solution of (6.65) can be derived from setting its derivative to zero

t τ =t-Lt+1 β t-τ W (n) τ ⊤ P (n) τ,m ŷ(n) τ,m ⊤ = t τ =t-Lt+1 β t-τ W (n) τ ⊤ P (n) τ,m W (n) τ u (n) t,m ⊤ . (6.66)
Instead of solving (6.66) directly, we propose a more elegant recursive way to obtain u

(n)
t,m as follows. First, let us denote the left hand side of (6.66) by d (n) t,m , and

S (n) t,m = t τ =t-Lt+1 β t-τ W (n) τ ⊤ P (n) τ,m W (n) τ . (6.67) 6.3.

TENSOR TRACKING WITH SPARSE OUTLIERS

Accordingly, (6.66) becomes

S (n) t,m u (n) t,m ⊤ = d (n) t,m . (6.68) Interestingly, both d (n)
t,m and S

t,m can be updated recursively:

d (n) t,m = βd (n) t-1,m + W (n) t ⊤ P (n) t,m ỹ(n) t,m ⊤ , (6.69)
S (n) t,m = βS (n) t-1,m + W (n) t ⊤ P (n) t,m W (n) t . (6.70)
where

W (n) t = W (n) t ⊤ W (n) t-Lt ⊤ ⊤ , (6.71)
ỹ(n) t,m = ŷ(n) t,m ŷ(n) t-Lt,m , (6.72)
P (n) t,m =   P (n) t,m 0 0 -β Lt P (n) t-Lt,m   . (6.73)
Therefore, we can rewrite (

= βS (n) t-1,m u (n) t-1,m ⊤ + W (n) t ⊤ P (n) t,m ỹ(n) t,m ⊤ = S (n) t,m u (n) t-1,m ⊤ + W (n) t ⊤ P (n) t,m ỹ(n) t,m ⊤ -W (n) t u (n) t-1,m ⊤ . 6.68) as S (n) t,m u (n) t,m ⊤ = βd (n) t-1,m + W (n) t ⊤ P (n) t,m ỹ(n) t,m ⊤
Multiplying both sides by S

(n) t,m -1 results in u (n) t,m = u (n) t-1,m + δỹ (n) t,m ⊤ V (n) t,m ⊤ , (6.75)
where

δỹ (n) t,m = P (n) t,m ỹ(n) t,m ⊤ -W (n) t u (n) t-1,m ⊤ , (6.76a)
V (n) t,m = S (n) t,m -1 W (n) t ⊤ . (6.76b)
Collecting all rows u

(n)

t,m together, m = 1, 2, . . . , I n , a simplified version of (6.75) for updating the whole factor U (n) t can be given by 9

U (n) t = U (n) t-1 + ∆ Y (n) t V (n) t ⊤ , (6.77)
9 To enable the recursive rules of (6.75) and (6.77), S

0,m and S

(n) 0 can be initialized by δIr where δ > 0, for n = 1, 2, . . . , N .

where

S (n) t = βS (n) t-1 + W (n) t ⊤ W (n) t , (6.78a)
V (n) t = S (n) t -1 W (n) t ⊤ , (6.78b)
∆ Y (n) t = P (n) t ⊛ Y (n) t -U (n) t-1 W (n) t ⊤ , (6.78c) with Y
(n) t = Y (n) t Y (n) t-Lt,m .
In this way, we can skip several operations and save a memory storage of O N -1

n=1 (I n -1)(I n r + r 2 ) . Specifically, the cost of computing (6.78a) is O r 2 N -1 i=1,i̸ =n I i . The computation of (6.78b) also requires a cost of O r 2 N -1 i=1,i̸ =n I i be- cause S (n) t
is of size r × r and its inverse computation is not expensive and independent of the tensor dimension. The error matrix ∆ Y (n) t in (6.78c) can be derived from Step 1 by reshaping the residual vector P t (y t -o t -H t-1 u t ). The most expensive step is the product ∆ Y

(n) t V (n) t ⊤ which costs r N -1
i=1 I i flops while the addition operator in (6.77) requires only rI n flops. Therefore, the overall cost of updating

U (n) t in a naive way is O r N -1 i=1 I i . Note that ∆ Y (n) t V (n) t ⊤ can be divided into two parts Z (n) t = ∆ Y (n) t W (n) t and Z (n) t S (n) t -⊤ . Here, ∆ Y (n) t W (n) t
can be referred to as "matricized tensor times Khatri-Rao product" (MTTKRP) [START_REF] Bader | Efficient MATLAB computations with sparse and factored tensors[END_REF][START_REF] Phan | Fast alternating LS algorithms for high order CANDECOMP/PARAFAC tensor factorizations[END_REF]. Fortunately, Phan et al. in [START_REF] Phan | Fast alternating LS algorithms for high order CANDECOMP/PARAFAC tensor factorizations[END_REF] proposed a clever reorganization of MTTKRP which can accelerate the computation and reduce the overall cost of (6.77) to O r 2 N -1 i=1,i̸ =n I i .

Stage 3: Normalization and re-estimation of u

(N ) t (Optional)
In order to avoid numerical problems, we can perform the column-wise normalization on the updated factors {U

(n) t } N -1 n=1 .
In addition, given the already estimated factors, the weight vector u t in Step 1 can be re-updated to achieve a better estimation as follows

u (N ) t = H ⊤ t P t H t # H ⊤ t P t ŷt , (6.79)
where

H t = N -1 n=1 U (n)
t . This step is useful for the early stage of tracking and fast time-varying environments [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF][START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF][START_REF] Vandecappelle | Nonlinear least squares updating of the canonical polyadic decomposition[END_REF].

Extensions of the RACP algorithm

In the following, we present two simple modifications of RACP when smoothness and nonnegativity are imposed on the loading factors.

Smoothness Condition

In many applications, smoothness is a common assumption under which the underlying data or model is supposed to be smooth [START_REF] Simonoff | Smoothing Methods in Statistics[END_REF]. Here, we incorporate a smoothing regularization matrix 6.3. TENSOR TRACKING WITH SPARSE OUTLIERS on the loading factors to control the smoothness of the solution as well as to avoid biases and singular/ill-posed computation. Particularly, this regularization adds a small bias against large terms into the updating rules.

On the arrival of Y t , the outliers O t and the coefficient vector u t are derived from the following minimization:

O t , u (N ) t = argmin O,u O 1 + γ 2 Bu 2 2 , subject to P t ⊛ Y t -O -H t-1 × N u 2 F = 0, (6.80)
where

H t-1 = I N -1 n=1 × n U (n)
t-1 and γ > 0 is a small penalty parameter and B a chosen banded matrix. More concretely, the vector u t is obtained by minimizing the following problem:

u (N ) t = argmin u γ 2 Bu 2 2 + ρ 2 P t (y t -o -H t-1 u) 2 2 . (6.81)
Accordingly, we replace the update rule for u in (6.56) with

u i = H ⊤ t-1 P t H t-1 + γ ρ B ⊤ B # H ⊤ t-1 P t y t -o i . (6.82)
Instead of (6.65), the m-th row u

(n) t,m of U (n) t is derived from u (n) t,m = argmin u (n) m t τ =t-Lt+1 β t-τ P (n) τ,m ŷ(n) τ,m ⊤ -W (n) τ u (n) m ⊤ 2 2 + γ 2 B u (n) m ⊤ 2 2 , (6.83)
In particular, u

t,m is the solution of the following equation:

t τ =t-Lt+1 β t-τ W (n) τ ⊤ P (n) τ,m ŷ(n) τ,m ⊤ = t τ =t-Lt+1 β t-τ W (n) τ ⊤ P (n) τ,m W (n) τ + γ 2 B ⊤ B u (n) m ⊤ . (6.84)
Therefore, the recursive rule of (6.75) becomes

u (n) t,m = u (n) t-1,m + δỹ (n) t,m ⊤ V(n) t,m ⊤ , (6.85)
where

V(n) t,m = S (n) t,m + γ 2 B ⊤ B -1 W (n) t ⊤ . (6.86) 6.3.

TENSOR TRACKING WITH SPARSE OUTLIERS

Nonnegative Constraint

It is known that nonnegative tensor factorization (NTF) offers interesting properties, e.g., the resulting expression appears to be purely additive and the loading factors are "sparse" in general [START_REF] Cichocki | Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation[END_REF].

One of the simplest ways is to project the estimates (i.e., u

(N ) t and {U (n) t } N -1 n=1
) on their nonnegative orthant at the end of each step of RACP, as introduced by Nguyen et al. in [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF]. This approach offers a low complexity and yields a reasonable performance in some cases. However, it may not be optimal as well as guarantee the convergence in general. In this task, we aim to customize the updates of u

(N ) t and {U (n) t } N -1
n=1 for dealing with the nonnegativity at each time t.

In step 1, we particularly replace the exact LS solution (6.56) with the minimizer of the following nonnegative least-squares (NNLS) problem:

u i = argmin u P t y t -o i -H t-1 u 2 2 subject to [u] j ≥ 0 ∀j. (6.87)
Here, we can apply any provable NNLS algorithm for solving (6.87), the reader is referred to [339,[START_REF] Lawson | Solving Least Squares Problems[END_REF] for good surveys on numerical methods for NNLS. In this work, we adopt the widelyused algorithm of Lawson and Hanson [START_REF] Lawson | Solving Least Squares Problems[END_REF] which is implemented as the function lsqnonneg in MATLAB.

In step 2, the m-th row of U (n) t can be derived from minimizing the following constrained version of (6.65):

u (n) t,m = argmin u (n) m t τ =t-Lt+1 β t-τ P (n) τ,m ŷ(n) τ,m ⊤ -W (n) τ u (n) m ⊤ 2 2 , subject to u (n) m j ≥ 0 ∀j. (6.88)
To solve (6.88), we apply the projected gradient method (i.e., proximal gradient on indicator function [START_REF] Parikh | Proximal Algorithms[END_REF]). More concretely, the iterative procedure for updating u

(n)
t,m is given by 10

u j = I r - S (n) t,m S (n) t,m 2 u j-1 - d (n) t,m S (n) t,m 2 + , (6.89)
where j denotes the iteration index. We refer to this modification of RACP as NRACP. 10 Projected gradient descent has a form of uj = uj-1 -ηj∇ ft(uj-1) + , where ∇ ft u

(n) m = S (n) t,m u (n) m -d (n)
t,m . In practice, we can set the value of the step-size ηj to 1/L where L is the Lipschitz constant of ∇ ft u

(n) m . In this work, it is easy to indicate that L = S (n) t,m 2 .

TENSOR TRACKING WITH SPARSE OUTLIERS

and tracking tools in particular to study their convergence and asymptotic behavior. In this work, a novel theoretical approach is established to analyse the convergence behavior of RACP in stationary environments. We leave the convergence analysis of RACP under a nonstationary model where the tensor dictionary is time-varying to a future work. Assumption (A3) is also common, under which the index of missing entries is uniformly random. Moreover, with respect to the imputation of missing values and recovery of low-rank components, the uniform randomness allows the sequence of binary masks {P t } t≥1 to admit stable recovery [START_REF] Chatterjee | A deterministic theory of low rank matrix completion[END_REF]. The next two constraints of (A3) are fundamental conditions to prevent the underdetermined imputation problem [START_REF] Pimentel-Alarcón | A characterization of deterministic sampling patterns for low-rank matrix completion[END_REF][START_REF] Ashraphijuo | Deterministic and probabilistic conditions for finite completability of low-Tucker-Rank tensor[END_REF]. The last constraint of (A3) plays a similar role as the first one but accounting for sparse outliers. Assumption (A4) allows us to derive several nice results in the convergence analysis. In fact, the Hessian matrix of ft (•) w.r.t. each factor is already positive semidefinite, (A4) can be achieved with a good initialization D 0 or by simply adding a convex regularization term into l(•) or ft (•).

Main Results

Given the assumptions of (A1)-(A4), our main theoretical result can be stated in the following theorem: Theorem 4. Given (A1)-(A4), L t = t and let D t be the solution generated by Algorithm 1 at each time t. When t → ∞,

• f t (D t ) -ft (D t ) a.s. → 0; • ∇f t (D t ) a.s. → 0.
Accordingly, D t is almost surely a stationary point of f t (.) when t tends to infinity.

The proof of this theorem follows intermediately Proposition 11 and Lemmas 12 and 13, to be stated shortly. We detail their proofs in our appendix.

Lemma 11 (Key Properties). Given (A1)-(A4), L t = t, and denote the error function

e t := ft -f t . If {D t , O t , u t } ∞
t=1 is a sequence of variables generated by Algorithm 1, then Lemma 12 (Almost sure convergence). The sequence of { ft (D t )} ∞ t=1 converges almost surely as t → ∞. The sequence of objective values {f t (D t )} ∞ t=1 converges to the same limit of its surrogate { ft (D t )} ∞ t=1 , i.e.,

(a) Boundedness: {D t , O t , u t } ∞ t=1 are uniformly bounded; (b) Forward Monotonicity: ft (D t-1 ) ≥ ft (D t ); (c) Backward Monotonicity: ft-1 (D t-1 ) ≤ ft-1 (D t ); (d) Stability of Estimates: ∥D t -D t-1 ∥ F = O(1/t); (e
f t (D t ) → ft (D t ) a.s. (6.90)
Proof Sketch. We first prove that

∞ t=1 E δ t E ft+1 (D t+1 ) -ft (D t ) F t < ∞, (6.91)
where F t = {D τ , O τ , u τ } 0<τ ≤t records all past estimates of RACP at time t and the indicator function δ t is defined as

δ t ∆ =    1 if E ft+1 (D t+1 ) -ft (D t ) F t > 0, 0 otherwise. (6.92)
Thanks to the quasi-martingale convergence theorem [343, page 51], (6.91) implies that { ft (D t )} ∞ t=1 converges almost surely as t → ∞.

We next prove {f t (D t )} ∞ t=1 and { ft (D t )} ∞ t=1 converge to the same limit by showing

∞ t=1 ft (D t ) -f t (D t ) t + 1 < ∞. (6.93) Since ∞ t=1 1 t+1 = ∞ and e t (D t ) -e t-1 (D t-1 ) = O(1/t), we obtain ∞ t=1 ft (D t ) -f t (D t ) < ∞, or ft (D t ) → f t (D t ) a.s., (6.94)
thanks to [120, Lemma 3].

TENSOR TRACKING WITH SPARSE OUTLIERS

Lemma 13 (Local convergence). When t → ∞, D t converges almost surely to a stationary point of f∞ (.) = lim t→∞ ft (.):

∇ f∞ (D t ) → ∇f ∞ (D t ) → 0 a.s. (6.95)
Proof Sketch. We first indicate that

lim t→∞ tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) = 0, (6.96)
by showing

∞ t=1 tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) < ∞.
Next, we prove that the following inequality

tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 ≤ c 1 D t+1 -D t 2 F + c 2 tr (D -D t ) ⊤ ∇ ft+1 D t , (6.97)
holds for all D ∈ D where c 1 and c 2 are positive constants.

Then, we use proof by contradiction to indicate that

∇ f∞ (D ∞ ) ⊤ (D -D ∞ ) ≥ 0, ∀D ∈ D. (6.98)
Accordingly, D ∞ is a stationary point of f∞ (.).

In order to prove ∇ ft D t a.s.

→ ∇f t D t as t → ∞, we first exploit that f t (D + a t V) ≤ ft (D + a t V) ∀D, V ∈ D and a t , and then take its Taylor expansion at t → ∞ to yield

f ∞ D ∞ + tr a t V ⊤ ∇f ∞ D ∞ + o a t V ≤ f∞ D ∞ + tr a t V ⊤ ∇ f∞ D ∞ + o a t V . (6.99) As indicated in Lemma 12, f∞ D ∞ = f ∞ D ∞ and thus tr a t V ⊤ ∇f ∞ D ∞ ≤ tr a t V ⊤ ∇ f∞ D ∞ .
Since the above inequality must hold for all V ∈ D and a t , we obtain

∇ f∞ D ∞ = ∇f ∞ D ∞ . (6.100)
Together with (6.98), we can conclude that D ∞ is a stationary point of the objective function f t (.) as t → ∞.

PERFORMANCE EVALUATION

Discussions

Our analysis follows the same framework to derive the convergence of adaptive/incremental algorithms for online matrix/tensor factorization problems as in [25,29,[START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF][START_REF] Feng | Online robust PCA via stochastic optimization[END_REF][START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF]. Therefore, our main theoretical result is somehow similar to their results. However, there are several points that make our convergence analysis different from theirs.

First, [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF] is devoted to the problem of online dictionary learning and sparse coding. The authors dealt with a LASSO-like cost function and required a preliminary uniqueness condition on the sparse coding. The condition is important to ensure that the solution generated in the sparse coding stage is unique, and to derive the Lipschitz property of the cost function. Particularly, they suggested an elastic-net regularized term for enforcing the condition. Since the problem formulation of RTT is different, our convergence analysis does not involve such issues. Moreover, the missing data distinguishes our work from theirs.

The studies in [START_REF] Feng | Online robust PCA via stochastic optimization[END_REF] and [25] consider the problem of robust online PCA/subspace tracking which can handle data corruptions (i.e., outliers and/or missing entries). These studies are designed for tracking the time-variant subspace -an object different from ours -which leads to some differences from our analysis. In particular, their main goal is to develop provable algorithms for minimizing the expected cost function in an online manner, and then indicate that their algorithm converges to a stationary point or global optimum under certain conditions.

Our optimization, however, minimizes an exponential weighted cost function constructed on the latest data streams (i.e., tensor slices). Moreover, [START_REF] Feng | Online robust PCA via stochastic optimization[END_REF] does not require the solution derived from the subspace update stage necessarily optimal, but full column rank only at each time t

(see [START_REF] Feng | Online robust PCA via stochastic optimization[END_REF]Theorem 1]). However, it is a sufficient condition on which we highly leverage in our analysis. In addition, our object is a set of multiple loading factors, instead of a single subspace matrix as in [25,[START_REF] Feng | Online robust PCA via stochastic optimization[END_REF].

The studies most related to ours are those in [29,[START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], which also investigate the tensor tracking problem. However, they consider only outlier-free streaming tensors. By contrast, we here provide a more unified convergence analysis that is able to deal with both missing data and outliers. Also, our results are stronger than those of [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], being limited to the case of third-order streaming tensors with β = 1.

Performance Evaluation

In this section, we provide several experiments on both synthetic and real data to demonstrate the effectiveness of the proposed algorithms, ACP, ATD, and RACP. We also compare them with several state-of-the-art algorithms to provide practical evidences of their effectiveness and efficiency. All experiments are implemented on MATLAB a windows computer with an Intel 6.4. PERFORMANCE EVALUATION Core i5-8300H and 16GB of RAM.12

Performance of ACP

We assess the performance of ACP w.r.t. the following aspects: (i) impact of algorithm parameters on its tracking ability; (ii) performance of ACP in non-stationary and time-varying environments; (iii) effectiveness and efficiency of ACP as compared with other adaptive CP algorithms.

Experiment Setup

According to the setup of OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], a time-varying model for streaming tensors is constructed as follows.

At t = 0, the loading factor U

(n) t is generated at random whose entries are i.i.d. drawn from the Gaussian distribution N (0, 1). At time t > 0, U

(n) t ∈ R In×r is varied under the model U (n) t = U (n) t-1 Q t , (6.101)
where Q t ∈ R r×r is a rotation matrix to control the variation of U (n) between instances t and t -1, which is defined by

Q t =          I pt-1 0 0 0 0 cos(α t ) -sin(α t ) 0 0 sin(α t ) cos(α t ) 0 0 0 0 I r-pt-1          , (6.102)
where p t = mod(t + r -2, r -1) + 1 and α t is the rotation angle. Specifically, the higher value of α t is, the faster the loading factor U (n) changes.

The t-th slice Y t with missing entries is then derived from the following model:

Y t = P t ⊛ U (n) t N -1 n=1 , u (N ) t + σN t , (6.103)
where P t is a binary mask tensor whose entries are generated randomly using the Bernoulli model with the probability ρ, i.e., ρ represents the missing density in the measurement; N t is a Gaussian noise tensor (with zero-mean, unit power entries) of the same size of Y t and the factor σ is to control the noise level; and the weight vector u (N ) t is a Gaussian random vector living on R r space. In order to evaluate estimation accuracy, we measure the relative error (RE) metric defined by

RE(A tr , A es ) = ∥A tr -A es ∥ F ∥A tr ∥ F , (6.104)
where A tr (resp. A es ) refers to the ground truth (resp. estimation) 13 .

Effect of Forgetting Factor β

The choice of β plays a central role in how effective and efficient ACP can be in nonstationary environments. In order to investigate the effect of the forgetting factor, we vary the value of β from 0 to 1 and measure estimation accuracy of ACP in different tests with regard to the rotational angle α. Fig. 6.3 illustrates the experimental results of applying ACP to a synthetic 4-order tensor whose size is 20 × 20 × 20 × 500 and its rank r = 5. The noise level σ is set at 10 -3 , while the sketching parameter m is fixed at 10. It is clear that the optimal value of β depends not only on the rotation angle α, but also on the missing density ρ. When β increases from 0 to 1, the performance of ACP goes up first and then drops. As can be seen in Fig. 6.3

that the value of β should be around 0.5 for reasonable performance. Thus, we fix β = 0.5 in the next experiments for. It is worth noting that in stationary environments, we can set the value of β = 1 to achieve the best performance, please see Fig. 6.4 for an illustration. 13 Due to the permutation and scaling indeterminacy of the CP decomposition, we can find Ues which is matched with Utr from Ut, as follows: Ues = UtP ⊤ D -1 , where the permutation matrix P ∈ R r×r and the diagonal matrix D ∈ R r×r are derived from minimizing the optimization argmin D,P Ut -UtrDP

Asymptotic Convergence Behavior

We next illustrate the convergence behavior of ACP in terms of the variation ∥U t+1 -U t ∥ F and the objective value f t (U t ). We use the same 4-order tensor above but with 1000 tensor slices.

Two noise levels are considered (including σ = 0 and σ = 10 -3 ), while the missing density ρ is chosen among {10%, 30%, 50%}. The experiment results are shown as in Fig. 6.5. We can see that convergence results agree with those stated in the proof sketch of Lemma 9.

Noisy and Dynamic Environments

First, the robustness of ACP is investigated against the noise variance. We test ACP's tracking ability on the same static 4-order tensor above with different values of the noise level σ. Fig. 6.6

shows that the value of σ does not affect the convergence rate of ACP, but only its estimation error. Specifically, when we increase the noise level σ, the relative error (RE) between the ground truth and estimation goes up gradually, but towards an error bound.

Next, we use the same tensor, but the number of slices is double for illustrating the robustness of ACP against time-varying environments. In particular, the proposed algorithm is evaluated in two scenarios, including a slow time-varying case (i.e., α = π/360) and a fast time-varying case (i.e., α = π/45). Also, at time t = 600, we make an abrupt change in these models. In addition, the missing density ρ is chosen among {10%, 30%, 50%}.

Experimental results indicate that ACP is capable of tracking streaming tensors in dynamic environments, as shown in Fig. 6.7. In both scenarios, the relative error (RE) between the ground truth and estimation always converges towards a steady state error bound. The missing density ρ has only an influence on the convergence rate of ACP. Specifically, the lower the missing density ρ is, the faster ACP converges.

Evaluation of Effectiveness and Efficiency

To demonstrate the effectiveness and efficiency of our algorithm, we compare performance of ACP in terms of estimation accuracy and running time with the state-of-the-art adaptive CP decompositions for incomplete tensors, including OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], CP-PETRELS [START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF], TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF].

For a fair comparison, parameters of these algorithms are fine-tuned carefully to achieve good performance. Particularly, the forgetting factor λ is set at 0.7, 0.001, and 0.98, respectively, for OLSTEC, TeCPSGD and CP-PETRELS. Moreover, OLSTEC and TeCPSGD are also dependent on a regularization parameter µ which is set at 10 -3 and 10 -1 respectively.

Since these algorithms are capable of tracking 3-order tensors only, we use synthetic streaming tensors of size N × N × 1000 in this task. The noise level is fixed at σ = 10 -3 . Performance of these algorithms is evaluated on a small tensor 20 × 20 × 1000 and a big tensor 200 × 200 × 1000.

Results are shown in Figs. 6.8 and 6.9. We can see that OLSTEC and ACP provide comparative estimation accuracy. In terms of running time, ACP is several times faster than OLSTEC, especially in big tensor tests. TeCPSGD is a fast adaptive algorithm, but yields lower estimation accuracy as compared to ACP and OLSTEC, while CP-PETRELS gives the worst accuracy as well as running time.

Performance of ATD

The following experiments will evaluate the ability of ATD for the problem of tensor tracking.

Experimental Setup

Follow the setup above, the incomplete slice Y t at time t is generated randomly using the following model:

Y t = P t ⊛ G t ; U (n) t N -1 t=1 , u (N ) t + σN t , (6.105)
where the loading factor U (n) t

and the core tensor G t are updated by the following rules where

U (n) t = U (n) t-1 + εN (n) t and G t = G t-1 + εV t , (6.106) 6.4.
U (n) 0 , N (n) t
∈ R In×rn and V t ∈ R r 1 ×r 2 ו••×r N are the Gaussian noises whose entries are distributed i.i.d from N (0, 1) and the time-varying factor ε is to control their variation.

Besides the relative error (RE) metric, we also use the subspace estimation performance (SEP) [START_REF] Linh-Trung | Lowcomplexity adaptive algorithms for robust subspace tracking[END_REF] metric to evaluate the subspace estimation accuracy, which is defined by

SEP(U tr , U es ) = tr U # es (I -U tr U # tr )U es tr U # es (U tr U # tr )U es , (6.107)
where U tr (resp. U es ) refers to the true loading factor (resp. estimated factor). The lower value of SEP is, the better accuracy the algorithm achieves.

Robustness of ATD

In order to demonstrate the robustness of ATD against data corruption, we change the number of missing entries in the measurement by varying the value of ρ and evaluate its performance on different noise levels. We also compare ATD with three well-known batch Tucker algorithms for tensor completion, including iHOOI [START_REF] Xu | Fast algorithms for higher-order singular value decomposition from incomplete data[END_REF], ALSaS [START_REF] Xu | Fast algorithms for higher-order singular value decomposition from incomplete data[END_REF], and WTucker [START_REF] Filipović | Tucker factorization with missing data with application to lown-rank tensor completion[END_REF]. These algorithms are iterative-based, so their procedure will be stopped when the accuracy tolerance tol or the maximum iteration ITER max has been met. For convergence guarantee, we fix the value of tol at 10 -4 , while the value of ITER max is set at 500, 500, and 100, respectively, for iHOOI, ALSaS and WTucker. For ATD, the forgetting factor λ is fixed at 0.7 in the following experiments.

In this task, we use a static tensor of size 20×20×20×500 (i.e., the time-varying factor ε = 0)

whose core is generated at random from a Gaussian distribution of zero-mean and unit variance.

Under the Tucker model with r TD = [START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF][START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF][START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF][START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF], performance of ATD on the tensor is illustrated 6.4. PERFORMANCE EVALUATION in Fig. 6.10. Results show that ATD can successfully track the multilinear low-rank model in all test cases. Similar to ACP, the missing density ρ has influence only on the convergence rate of ATD, i.e., the higher the value of ρ is, the slower ATD converges. Performance comparison among Tucker algorithms is reported statistically in Tab. 6.1 and shown in Fig. 6.11. Results

Missing ρ = 50% ρ = 70% ρ = 50% ρ = 70% Metric RE(X ) SEP(U) Time(s) RE(X ) SEP(U) Time(s) RE(X ) SEP(U) Time(s) RE(X ) SEP(U)
indicate that ATD is the fastest algorithm, much faster than the state-of-the-art algorithms.

For instance, when dealing with the case of 50% missing observations and r TD = [3, 3, 3, 3], the running time of ATD is only 2.51 seconds and 35 times faster than the second-fastest algorithm, iHOOI. Moreover, ATD always provides good estimation accuracy in terms of both SEP metric and RE metric as compared to that of the batch algorithms.

Tracking Ability in Dynamic Environments

We continue to investigate the tracking ability of ATD in nonstationary and time-varying environments by changing the time-varying factor ε in the range [10 -4 , 10 -1 ]. We use the same tensor dimensions as in the previous task. Also, we create a significant subspace change at time t = 300 to see how fast ATD can converge. Fig. 6.12 shows the convergence behavior of ATD versus the time-varying factor ε. We can see that the convergence rate of ATD is not affected by ε but only its estimation error.

Orthogonality Constraint

In practice, Tucker decomposition is often considered with orthogonality constraints on the loading factors. The unconstrained ATD can be recast into an orthogonal ATD while retaining the equivalent approximation error. To demonstrate this point, we set up a time-varying scenario and compare the performance of ATD and ATD with the orthogonalization step, called ATD-O.

Fig. 6.12 indicates that the convergence rate of ATD-O is slightly better than that of the unconstrained ATD, but both yield the same error floor. Due to space limitation, we here omit results

with ATD-O and presents only those of ATD.

Real Data

To demonstrate the effectiveness of our algorithms on real datasets, we consider two related applications: video completion and multichannel EEG analysis. We first investigate the effect of the forgetting factor λ on the reconstruction performance of the two proposed algorithms for video completion. Particularly, the value of λ and the missing ratio ρ are varied from 0.1 to 0.9. The CP rank and Tucker rank are set at 10 and [10, 10, 10],

Video

respectively. Experimental results from Fig. 6.14 indicate that the performance of ACP and ATD is not much affected by the forgetting factor. For this task, we therefore keep the value of λ at 0.5 as in previous experiments on synthetic data. We next compare our algorithms with OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF] and CP-PETRELS [START_REF] Minh-Chinh | Adaptive PARAFAC decomposition for third-order tensor completion[END_REF]. We set the value of λ at 0.7, 0.001 and 0.999, respectively, for OLSTEC, TeCPSGD and CP-PETRELS. Besides, OLSTEC and TeCPSGD are also depended on the regularization parameter µ which value is fixed at 0.1. Performance of these algorithms is shown statistically in Tab. 7.1 and graphically in Fig. 6.15. We can see that ATD outperforms adaptive CP algorithms in almost all tests. ACP also provides reasonable estimation accuracy on these data as compared to others. CP-PETRELS seems to fail to track video background and thus recovers missing data unsuccessfully. With respect to the running time, experimental results indicate that ACP is the fastest adaptive tensor decompositions.

Multichannel EEG Analysis. We follow the experimental framework in [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF][START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF] to illustrate the use of ACP for analyzing multichannel EEG signals. In this task, we use a public EEG dataset collected on 14 subjects during the stimulation of hands 15 . The EEG signals are recorded using a system of 64 channels (electrodes) and we have 28 measurements per subject in total.

We construct three-order EEG tensor of measurement × channel × time-frequency by taking continuous wavelet transform to each EEG channel. Note that, the resulting time-frequency representations are reshaped into vectors of length 4392 and hence being streamed. In a nutshell, we have the EEG tensor whose size is 28×64×4392 and its rank is set at 3 as provided in [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF][START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF].

At each time, data of 20 (and 40) channels are supposed to be discarded randomly for our missing observation purpose. We evaluate the performance of ACP in a comparison with the adaptive NL-PETRELS algorithm in [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF] and the batch CP-WOPT algorithm in [START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF]. To have a good initialization for NL-PETRELS, the 1500 first slices are used to construct the training tensor. Also, the forgetting factor λ is set at 0.999. By contrast, ACP is not as sensitive to initialization conditions, so it is initialized at random. We consider results obtained by using the batch algorithm as our ground truth.

Under the CP model with r CP = 3, taking the tensor decomposition to the EEG tensor results in three loading factors A ∈ R 28×3 , B ∈ R 64×3 and C ∈ R 4392×3 corresponding to, respectively, the measurement, channel and time-frequency modes. Fig. 6.16 illustrates the estimation of A, B and C using CP-WOPT, NL-PETRELS and ACP. In particular, we use bar plots and 3D head plots to represent the column vectors of A and B, while the time-frequency representations corresponding to the columns of C are plotted as matrices. We can see from Fig. 6.16 that both adaptive algorithms are capable of tracking three EEG loading factors. Indeed, our ACP provides a slightly better estimation as compared to that of CP-WOPT. However, in the presence of highly incomplete observations (e.g. 40 channels are missing), NL-PETRELS fails to estimate the EEG loading factors while our ACP algorithms still works well, as shown in Fig 6 .17.

Performance of RACP

We here provide several experiments on both synthetic and real data to demonstrate the effectiveness of RACP and its variant. In particular, the performance of our method is evaluated in comparison with the-state-of-the-art algorithms with respect to the following aspects: (i) im- pact of outliers, (ii) impact of missing data, and (iii) tracking ability in noisy and time-varying environments.

Experiment Setup

At t = 0, the loading factor U (n) 0 ∈ R In×r , n = 1, 2, . . . , N is randomly initialized whose entries are i.i.d. from a normal distribution N (0, 1). When t ≥ 1, U (n) t is varied according to the following model: where

U (n) t = U (n) t-1 + ϵN (n) t , (6.108)
N (n) t
is a Gaussian noise matrix (with zero-mean and unit-variance), and ϵ is a positive time-varying factor used to control the variation of U (n) between t and t -1.

The t-th slice Y t is then generated under the data model

Y t = P t ⊛ U (n) t N -1 n=1 , u (N ) t + O t + N t , (6.109)
where P t is a binary observation mask according to a Bernoulli distribution with probability of observing data 1 -ω miss , N t is a Gaussian noise tensor with i.i.d. entries N (0, σ a sparse outlier tensor whose entries are drawn uniformly from the range [0, A outlier ] and the indices of outliers also follow a Bernoulli distribution with probability ω outlier , and u t ∈ R r×1 is a standard normal random vector.

Robustness of RACP

We first investigate the robustness of RACP against gross data corruptions. Specifically, we change the density of outliers and missing data, and then measure the relative error between the ground truth and RACP's estimation.

In this task, we use a synthetic 4 th -order streaming tensor of size 20 × 20 × 20 × 1000 and the CP rank is set at r = 2 and r = 5. The noise level σ n and the time-varying factor ϵ are fixed at 10 -3 and 10 -2 , respectively. We consider the case where the underlying data is corrupted by strong outliers with A outlier = 10. The fraction of outliers (ω outlier ) and missing data (ω miss ) ]. An abrupt change is created at t = 600 to assess how fast RACP converges. We can see from Fig. 7.12 that RACP's convergence rate is not much affected by the value of ϵ but its estimation accuracy.

To demonstrate the effectiveness of the proposed algorithm, we compare the performance of RACP with the state-of-the-art adaptive CP decompositions, including TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], OL-STEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], and ACP [29]. To have a fair comparison, algorithm parameters are set by default as suggested by their authors. These algorithms are dependent on a forgetting factor; we set its value at 0.7, 0.001, and 0.5 for OLSTEC, TeCPSGD, and ACP, respectively. The penalty parameter is set at 10 -3 and 10 -1 for OLSTEC and TeCPSGD, respectively.

Since OLSTEC and TeCPSGD are only capable of tracking third-order streaming tensors, we here use a synthetic streaming tensor of size 20 × 20 × 1000 and its rank is fixed at 5. The noise level and time-varying factor are both kept at 10 -2 . Performance comparison results are shown in Figs. 6.20 and 6.21.

Fig. 6.20 illustrates the impact of the outlier intensity on the performance of the four adaptive CP algorithms in the presence of 10% missing data and 20% outliers. When the outlier intensity is small, all algorithms could track the underlying tensor model over time, as shown in Fig. 6.20(a).

Indeed, TeCPSGD yielded a worse estimation than the three remaining adaptive CP algorithms.

In the presence of strong outliers, the state-of-the-art adaptive CP algorithms failed to update the tensor basis and recover the corrupted tensor slice. By contrast, our RACP algorithm still worked well, as shown in Fig. 6.20(b). Fig. 6.21 illustrates the impact of the outlier density on the performance of RACP against the three adaptive CP algorithms when the missing density ω miss = 10% and outlier intensity A outlier = 10. We can see that RACP outperformed OLSTEC, TeCPSGD, and ACP in all testing cases. Similar to the case study of strong outliers, the state-

(n) t = U (n) t-1 + ϵN (n) t
where N (n) t is also an i.i.d. uniform random matrix from 0 to 1. The parameter specifications are kept as in the previous experiment.

Results are illustrated in Fig. 6. [START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF]. We can see that the proposed RACP algorithm still tracks successfully the loading factors along the time while the state-of-the-art CP algorithms failed.

Experimental results in Figs. 6.20, 6.21, and 6.22 suggest that the outlier rejection step (e.g.

Step 1 in RACP) using the ADMM solver plays an important role in the tracking process when observations are corrupted by sparse outliers. Therefore, we next evaluate the effectiveness of the proposed outlier rejection by applying the ADMM solver to other trackers: TeCPSGD and OLSTEC. We here reuse the experiment setup above and create an abrupt change at t = 600. We can see from Fig. 6.23 that the combination of the ADMM solver and OLSTEC resulted in the best convergence rate and estimation accuracy. This is probably due to the effectiveness of the second-order estimator in slowly time-varying environments. Our RACP provided a reasonable performance compared to that of OLSTEC, while TeCPSGD tracker did not work well. It should note that OLSTEC is designed for only 3 rd -order streaming tensors and its computational complexity is high indeed. Our tracker is much faster and capable of dealing with higher-order streaming tensors. We refer the readers to our companion work in [29] for further comparisons of ACP against TeSGD and OLSTEC.

Finally, we conduct a performance comparison between the original RACP and its variant in which the step of re-updating P t defined as in (6.61) is used. We reuse the two rank-5 tensors of size 20 × 20 × 20 × 1000 and 20 × 20 × 20 × 20 × 1000. The fraction of missing entries is fixed at 10%. We set the outlier density and intensity to 10% and 10, respectively. The noise and time-varying factors are kept at 10 -2 and an abrupt change at t = 600 is also created as in previous experiments. The results are illustrated in Fig. 6.24. As can be seen the outlier rejection mechanism can help improve the convergence rate of RACP.

Nonnegative RACP

We reuse the experiment setup in Section 6.4.3.1, but the time variation of U (n) ⪰ 0 is modified as

U (n) t = abs U (n) t-1 + ϵN (n) t , (6.110)
where abs(•) denotes the absolute value, N

(n) t is a Gaussian noise matrix with i.i.d. entries, and ϵ is to control the variation. Next, we study the robustness of NRACP against the noise variance in comparison with NSOAP [START_REF] Dung | Second-order optimization based adaptive PARAFAC decomposition of three-way tensors[END_REF] and NsTEF [START_REF] Vigneron | Non-negative sub-tensor ensemble factorization (NsTEF) algorithm. A new incremental tensor factorization for large datasets[END_REF]. Since both two algorithms are only feasible for third-order tensors without corruptions (outliers and missing values), we use a synthetic outlier-free tensor of size 50 × 50 × 1000 and rank 5 for this task. The time-varying factor ϵ is set at 10 -3 . Performance comparison results are illustrated in Fig. 6.26. At a low SNR, NSOAP provided a better estimation accuracy than NRACP and NsTEF. However, the proposed NRACP outperformed NSOAP and NsTEF at the high SNR, see Fig. 6.26(b). In the presence of abrupt changes, the convergence rate of NRACP was fast while NSOAP and NsTEF failed to track the change.

Real Datasets

To demonstrate the use of RACP with real-world datasets, we consider the following tasks: (i) tracking the online low-rank approximation of real-world data streams, (ii) multichannel EEG analysis, and (iii) video background modeling and foreground detection. Please see Tab. 6.3 for a summary of real datasets used in this paper. domly generated with probabilities ω miss and ω outlier , respectively. Outlier's values are drawn uniformly from the range [0, max(X )] where max(X ) is the largest absolute value in the underlying data X . In this experiment, we choose the value of ω miss and ω outlier among the range {5%, 10%, 20%, 40%}. As the true rank is unknown, we first vary its value from 2 to 10 and then (5%,5%) (10%,10%) (20%,20%) (40%,40%) 10 -3 choose the "best" one based on the averaged reconstruction error, see Fig. 6.27(a) for an example.

We compare the performance of RACP against the two adaptive CP algorithms TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF] and OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF]. Both algorithms are dependent on the forgetting factor λ, and its value is set at 0.98, 0.001, and 0.7, respectively. The penalty parameter µ is set at 1 for both TeCPSGD and OLSTEC. The experimental result in Fig. In what follows, we consider two common problems in multichannel EEG analysis: (i) incomplete multichannel EEG analysis from partial observations and (ii) anomaly EEG detection.

Incomplete Multichannel EEG Analysis: Here, we use the ERPWAVELAB dataset and follow the same experimental setup in [29,[START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF][START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF] to demonstrate the use of RACP with real EEG 6.4. PERFORMANCE EVALUATION at each time are supposed to be missing at random. As suggested in [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF][START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF], we set the tensor rank at r = 3. Performance of RACP is compared with two adaptive CP algorithms NL-PETRELS [START_REF] Linh-Trung | A non-linear tensor tracking algorithm for analysis of incomplete multi-channel EEG data[END_REF] and ACP [29]. We fix the forgetting factor λ at 0.999 and 0.5 for NL-PETRELS and ACP, respectively. As NL-PETRELS requires a warm start, we run the batch CP-WOPT algorithm [START_REF] Acar | Scalable tensor factorizations for incomplete data[END_REF] with the first 1500 tensor slices. Meanwhile, we use random initialization for ACP Anomaly EEG Detection: We demonstrate the use of RACP to detect abnormal activities in the brain (i.e., epileptic seizures) with the epileptic EEG dataset. Here, we adopt a simple but effective way to predict abnormalities in multidimensional data streams [START_REF] Sun | Incremental tensor analysis: Theory and applications[END_REF]. In particular, we model the abnormality of a tensor (streaming) slice Y t by its recovery error

e t = P t ⊛ Y t -Y t N n=1 × n U (n) t U (n) t # F Y t F , (6.111)
where {U

(n) t } N n=1 is the set of solutions generated by RACP at time t. It is also worth noting that the error e t is relatively proportional to the norm of the outlier O t . We label Y t based on

where {e} Lt denotes the set of e τ with t -L t < τ ≤ t.

We follow the method in our companion work on epileptic spike detection [START_REF] Thanh | Multi-channel EEG epileptic spike detection by a new method of tensor decomposition[END_REF] Background Modeling. We first measure the video background modeling ability of RACP in comparison with a robust subspace tracking algorithm PETRELS-ADMM [25], and two adap- tive CP algorithms (TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF] and OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF]). These algorithms are dependent on the forgetting factor λ, and its value is set at 0.98, 0.001, and 0.7, respectively. The penalty parameter µ is set at 0.1 for both TeCPSGD and OLSTEC. The CP rank and subspace rank are set at 10.

We consider the scenario where 50% of pixels are supposed to be missing at random. Experimental results are illustrated in Fig. 6.33. As we can see that the two robust algorithms PETRELS-ADMM and RACP were able to recover the video background. Particularly, the proposed RACP provided slightly better estimation than PETRELS-ADMM. The two adaptive CP algorithms TeCPSGD and OLSTEC seem to have failed when the video frame contains moving objects, probably because they do not account for sparse outliers.

Foreground Detection. Next, we investigate the ability of RACP in video foreground detection. We also compare the performance of RACP with three notable foreground detection algorithms, including GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF], OSTD [START_REF] Sobral | Online stochastic tensor decomposition for background subtraction in multispectral video sequences[END_REF] and PETRELS-ADMM [25]. To have a fair comparison, algorithm parameters are set by default as suggested by their authors. Particularly, the penalty parameter ρ and constant step-size scale C are, respectively, set at 1.8 and 2 in GRASTA. The forgetting factor in PETRELS-ADMM is fixed at λ = 0.98, while OSTD is a parameter-free algorithm. As can be seen from Fig. 6.34 that RACP was capable of detecting moving objects in video streams and provided a competitive performance as compared to GRASTA, OSTD, and PETRELS-ADMM.

Conclusions

In this chapter, we have proposed three new low-complexity algorithms (including ACP, ATD, and RACP) for adaptive decomposition of higher-order incomplete and streaming tensors. First, developed based on CP decomposition, ACP estimates a multilinear LRA of streaming tensors 6.6. APPENDIX Our analysis follows the same framework to derive the asymptotic convergence of adaptive algorithms for problems of online matrix and tensor factorization [25,[START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF][START_REF] Feng | Online robust PCA via stochastic optimization[END_REF]. In particular, the convergence analysis contains three main stages: (I) we show that the solutions {U t , u t } ∞ t=1 6.6. APPENDIX

Original Frame RACP PETRELS-ADMM GRASTA OSTD t=1 and its surrogate {g t (U t )} ∞ t=1 converge to the same limit, i.e., g t (U t ) → f t (U t ) a.s. Accordingly, {U t } ∞ t=1 converges to a stationary point of f t (U ), i.e. ∇f t (U t ) t→∞ -→ 0. Proof. We first note that ACP begins with full column rank and bounded factors {U

(n) 0 } N n=1 .
6.6. APPENDIX

The recursive rule for updating U

(n)

k+1 is given by

U (n) k+1 = U (n) k + ∆Y (n) k+1 V (n) k+1 ⊤ , (6.118)
where

∆Y (n) k+1 = P (n) k+1 ⊛ Y (n) k+1 -U (n) τ W (n) k+1
⊤ , (6.119a)

V (n) k+1 = S (n) k+1 -1 , (6.119b)
S (n) k+1 = βS (n) k + W (n) k+1 ⊤ W (n) k+1 , (6.119c)
W (n) k+1 = N i=1,i̸ =n U (i) k ⊙ u ⊤ k+1 , (6.119d)
Since {U

(n) k } N n=1 are assumed to be bounded and u k+1 is bounded, we obtain that W k+1 can be recursively expressed by

S (n) k+1 = λS (n) k + i w i w ⊤ i , (6.120)
where w i is the i-th row of W where σ max (M) and σ min (M) are the largest and smallest singular value of M, respectively.

Accordingly, we obtain

V (n) k+1 F ≤ √ r CP V (n) k+1 2 = √ r CP σ min S (n) k+1 . (6.122)
The lower bound on the minimum singular value of S

(n)

k+1 is specified by the following proposition.

Proposition 13 (Theorem 1 [START_REF] Bunch | Rank-one modification of the symmetric eigenproblem[END_REF] and Theorem 2.1 [START_REF] Cheng | The bounds of the smallest and largest eigenvalues for rank-one modification of the Hermitian eigenvalue problem[END_REF]). Let A be an r × r symmetric matrix with positive eigenvalues σ

1 (A) ≥ σ 2 (A) ≥ • • • ≥ σ r (A) > 0.
If w is an rdimensional column vector and  = A + ww ⊤ , we always have

σ r (A) ≤ σ r ( Â) ≤ σ r-1 (A) ≤ σ r-1 ( Â) ≤ • • • ≤ σ 1 (A) ≤ σ 1 ( Â) ≤ σ 1 (A) + ∥w∥ 2 2 .
(6.123) 6.6. APPENDIX Accordingly, we have

σ min S (n) k+1 ≥ λσ min S (n) k ≥ λ 2 σ min S (n) k-1 ≥ • • • ≥ λ k+1 σ min (S (n) 0 ) = λ k+1 δ n ≥ δ n . (6.124)
The last inequality is when the forgetting factor λ = 1. As a result, we obtain

V (n) k+1 F ≤ √ r CP δ -1 n < +∞. (6.125) It implies that V (n)
k+1 is bounded. Therefore, U k+1 is bounded, thanks to the rule (6.118).

Proposition 14. The surrogate g t (.) is a Lipschitz function.

Proof. First, we exploit that g t+1 U

(n) t+1 ≤ g t+1 U (n) t ∀ t due to U (n) t+1 = argmin g t+1 U (n)
and hence

g t U (n) t+1 -g t U (n) t = g t U (n) t+1 -g t+1 U (n) t+1 + g t+1 U (n) t+1 -g t U (n) t ≤ g t U (n) t+1 -g t+1 U (n) t+1 -g t U (n) t -g t+1 U (n) t ∆ = d t U (n) t+1 -d t U (n) t , (6.126)
where d t (U) = g t (U) -g t+1 (U). The derivative of d t (U (n) ) is then given by

∂d t U (n) ∂U (n) = U (n) A t t - A t+1 t + 1 + B t t - B t+1 t + 1 , (6.127)
where

A t = t τ =1 β t-τ W (n) τ ⊤ W (n) τ , B t = t τ =1 β t-τ P (n) τ ⊛ X (n) τ W (n)
τ . Accordingly, we have

∂d t U (n) ∂U (n) F ≤ U (n) F A t t - A t+1 t + 1 F + B t t - B t+1 t + 1 F , (6.128)
thanks to the following inequality ∥MN∥ F ≤ ∥M∥ F ∥N∥ F for all M, N. It implies that the

function d t (U (n) ) is Lipschitz, i.e., g t U (n) t+1 -g t U (n) t ≤ d t U (n) t+1 -d t U (n) t ≤ c n U (n) t+1 -U (n) t F , (6.129)
where the Lipschitz constant

c n = O(1/t) is given by c n = κ At t -A t+1 t+1 F + Bt t -B t+1 t+1 F , where ∥U (n) ∥ F ≤ κ is the upper bound for ∥U (n) ∥ F .
In parallel, the surrogate g t (U ) is a multi-convex function because of its quadratic form. It 6.6. APPENDIX is therefore that

g t U (n) t+1 -g t U (n) t ≥ m n U (n) t+1 -U (n) t 2 F , (6.130)
where m n is a positive number. From (6.129) and (6.130), we obtain the following nice corollary:

Corollary 3. The asymptotic variation of U t is given by

U (n) t+1 -U (n) t F = O 1/t . ( 6
where {F t } t>0 is the filtration of the past estimations at time instant t.

Proof. We begin with the expression

g t+1 (U t ) = 1 t + 1 t+1 τ =1 β t+1-k l(U t , P τ , Y τ , u τ ) = 1 t + 1 l(U t , P t+1 , Y t+1 , u t+1 ) + βtg t (U t ) = l(U t , P t+1 , Y t+1 , u t+1 ) t + 1 + t(β -1) t + 1 g t (U t ) + t t + 1 g t (U t ). (6.133)
6.6. APPENDIX where ℓ(U , P, Y) = min u l(U , P, Y, u). We then have

g t (U t ) -f t (U t ) t + 1 = g t (U t ) - t t + 1 g t (U t ) - f t (U t ) t + 1 = g t (U t ) -g t+1 (U t+1 ) + l(U t , P t+1 , Y t+1 , u t+1 ) -f t (U t ) t + 1 + g t+1 (U t+1 ) -g t+1 (U t ) ≤0 + t(β -1) t + 1 g t (U t ) ≤0 ≤ g t (U t ) -g t+1 (U t+1 ) + l(U t , P t+1 , Y t+1 , u t+1 ) -f t (U t ) t + 1 , (6.134)
because 0 < β ≤ 1 and g t+1 (U t+1 ) ≤ g t+1 (U t ) for all t due to U t+1 = argmin g t+1 (U ).

Moreover, we know that u t+1 = argmin u l(U t , P t+1 , Y t+1 , u), so

ℓ(U t , P t+1 , Y t+1 ) = l(U t , P t+1 , Y t+1 , u t+1 ).
Accordingly, we obtain the following inequality

g t+1 (U t+1 ) -g t (U t ) ≤ ℓ(U t , P t+1 , Y t+1 ) -f t (U t ) t + 1 - g t (U t ) -f t (U t ) t + 1 . (6.135)
Moreover f t (U t ) ≤ g t (U t ) for all t, we obtain

g t+1 (U t+1 ) -g t (U t ) ≤ ℓ(U t , P t+1 , Y t+1 ) -f t (U t ) t + 1 . (6.136)
Taking the expectation of (6.136) conditioned by F t results in

E g t+1 (U t+1 ) -g t (U t )|F t ≤ f (U t ) -f t (U t ) t + 1 , (6.137)
where f (U ) be the expected cost function, i.e., f (U ) = lim t→∞ f t (U ) and E ℓ(U , P t+1 , Y t+1 ) = f (U ), for all U . Now, let us define the indicator function δ t as follows

δ t ∆ =    1 if E g t+1 (U t+1 ) -g t (U t )|F t > 0, 0 otherwise. (6.138)
Accordingly, we have We then consider the convergence of the sum ∞ t=1 1 √ t(t+1) . In particular, the Cauchy-MacLaurin integral test [START_REF] Knopp | Theory and Application of Infinite Series[END_REF] is applied for examining the convergence, that is,

E δ t E g t+1 (U t+1 ) -g t (U t )|F t ≤ E √ t f (U t ) -f t (U t ) 1 √ t(t + 1) . ( 6
+∞ t=1 1 √ t(t+1) dt = π 4 < ∞. Accordingly, 1 √ t(t+1) t>0
converges. Therefore, we obtain We complete the proof.

∞ t=1 E δE g t+1 (U t+1 ) -g t (U t )|F t < ∞. ( 6

Stage III

The last stage contains two main steps: (i) we first indicate that the empirical cost function f t (U ) is not only continuously differentiable, but also Lipschitz; (ii) we then prove {f t (U t )} ∞ t=1 and {g t (U t )} ∞ t=1 converge to the same limit. As a result, the derivative of f t (U ) equals to that of g t (U ) when t → ∞, thanks to the first-order Taylor approximation. Since U t is the minimizer of g t (U ), the derivative ∇f t (U ) → 0 a.s.

To begin with, we provide the following proposition which is a corollary of Theorem 4.1 in [START_REF] Bonnans | Optimization problems with perturbations: A guided tour[END_REF]:

Proposition 16. Consider a continuous function f : V × U → R. Suppose that ∀u ∈ U, the function f (., u) is differentiable and ∇ v f (v, u) is continuous on V × U. If g(v) be the function derived from g(v) = min u∈U f (v, u), then g(v) is also differentiable. In addition, if u * = argmin u∈U f (v, u) be unique, ∇g(v) = ∇ v f (v, u * ), ∀v ∈ V.
Proof. Its proof is already provided in [START_REF] Bonnans | Optimization problems with perturbations: A guided tour[END_REF]Theorem 4.1].

Accordingly, we derive the following proposition to verify the differentiable property of ℓ(U , P t , X t ) at time t.

6.6. APPENDIX Corollary 4. Given an incomplete observation P t ⊛ X t and the past estimation U , let u * t be the minimizer of the summand l(U , P t , X t , u)

u * t = argmin ut∈R r P t ⊛ Y t -U (n) } N -1 n=1 , u (N ) t 2 F . (6.142)
We obtain that ℓ(U , P t , X t ) = min ut l(U , P t , X t , u t ) is a continuously differentiable function and its partial derivative w.r.t. U (n) is given by ∂ℓ(U , P t , X t )

∂U (n) = 2P (n) t ⊛ Y (n) t -U (n) W * t ⊤ W * t , (6.143)
where

W * t = N i=1,i̸ =n U (i) t-1 ⊙ (u * t ) ⊤ . (6.144)
As a result, the empirical cost function f t (U ) = 1 t β t-τ t τ =1 ℓ(U , P τ , Y τ ) is continuously differentiable. Applying the same augments in Proposition 14, we also have

∂ ft ∂U (n) F ≤ U (n) F Ā(n) t t - Ā(n) t+1 t + 1 F + B(n) t t - B(n) t+1 t + 1 F , (6.145)
where ft

U (n) = f t U (n) -f t+1 U (n) , Ā(n) t = t τ =1 β t-τ W * τ ⊤ W * τ , and
B(n) t = t τ =1 β t-τ P (n) τ ⊛ Y (n) τ W * τ .
All terms in the right side are bounded, the partial derivative ft (U ) w.r.t. U (n) is bounded and hence According to (6.174), we recall the following inequality

f t U (n) t+1 -f t U (n) t ≤ d n U (n) t+1 -U (n) t F , ( 6
g t (U t ) -f t (U t ) t + 1 ≤ g t (U t ) -g t+1 (U t+1 ) + ℓ(U t , P t+1 , Y t+1 ) -f t (U t ) t + 1 . (6.148)
To examine the convergence of the right side of (6.148), we exploit the following facts: (i) The convergence of E g t (U t )-g t+1 (U t+1 )|F t is already provided in Proposition 2, and (ii) The second term also converges, thanks to the convergence of E[f (U t ) -f t (U t )]/(t + 1) and 6.6. APPENDIX E[ℓ(U t , P, X )] = f (U t ) for all t.

Accordingly, we have that g

t (U t ) -f t (U t ) 1 t+1 converges ∞ t=0 g t (U t ) -f t (U t ) 1 t + 1 < ∞. (6.149)
Since both g t (U ) and f t (U ) are Lipschitz continuous, there always exist a constant L > 0 such that

g t+1 (U t+1 ) -f t+1 (U t+1 ) -g t (U t ) -f t (U t ) ≤ L U t+1 -U t F . (6.150)
In addition, the real sequence 1 t+1 t≥0 diverges, i.e., ∞

1 t+1 = +∞. It implies that ∞ t=0 g t (U t )- f t (U t ) < ∞, thanks to [351, Lemma A.5]. It results in g t (U t ) a.s → f t (U t ), t → ∞. t=0
In parallel, g t (U ) is the surrogate function of f t (U ), we always have

g t (U + a τ V) ≥ f t (U + a τ V), (6.151)
for all V and the nonnegative sequence {a τ }. For short, let us denote g t U (n) ∆ = g t (U ) and

f t U (n) ∆ = f t (U )
when the remaining loading factors are fixed. With respect to U (n) , the inequality (6.151) becomes

g t U (n) + a τ V (n) ≥ f t U (n) + a τ V (n) . (6.152)
Thanks to Taylor's theorem, taking the linear approximation of (6.180) yields

g t U (n) t + tr a τ V (n) ⊤ ∇g t U (n) t + o a τ V (n) ≥ f t U (n) t + tr a τ V (n) ⊤ ∇f t U (n) t + o a τ V (n) . (6.153) When t → ∞, we have g t U (n) t = f t U (n) t
as proved in Lemma 1 and hence

tr a τ V (n) ⊤ ∇g t U (n) t ≥ tr a τ V (n) ⊤ ∇f t U (n) t . (6.154)
Since the above inequality must hold for all V (n) and {a τ }, we obtain At each time t > 0, the outlier O t and the coefficient vector u t are derived from the minimization (7) in the main manuscript. Accordingly, we always have

∇g t U (n) t -∇f t U (n) t → 0, when t → ∞. ( 6
l D t-1 , P t , Y t , O t , u t ≤ l D t-1 , P t , Y t , 0, 0 . (6.156)
It is therefore that

∥O t ∥ 1 + ρ 2 P t ⊛ (Y t -O t -H t-1 × N u t ) 2 F ≤ ρ 2 P t ⊛ Y t 2 F . (6.157)
Due to the two facts that ∥M∥ F + ∥N∥ F ≥ ∥M -N∥ F ≥ ∥M∥ F -∥N∥ F , and

∥M∥ F ≤ ∥M∥ 1 [9],
we then obtain

O t F ≤ O t 1 ≤ ρ 2 P t ⊛ Y t 2 F ≤ ρ 2 M 2 x < ∞, (6.158)
P t H t-1 u t 2 ≤ 2 P t ⊛ Y t F + P t ⊛ O t F < ∞, (6.159)
where M x is the upper bound of ∥Y t ∥ F (see Assumption A1). Thanks to (6.158), O t is uniformly bound.

We indicate the bound of the solution u t and D t = U The base case: At t = 1, the matrix

H 0 = N n=1 U (n)
0 is then full rank, i.e., the null space of H 0 admits only 0 as a vector. Accordingly, u 1 is bounded, thanks to (6.159).

To indicate the bound of U (n) 1

for n = 1, 2, . . . , N , we show that each row u

(n) 1,m of U (n) 1
is bounded. We first obtain the following inequality

u (n) 1,m 2 ≤ u (n) 0,m 2 + P (n) 1,m x (n) 1,m ⊤ -W (n) 1 u (n) 0,m ⊤ 2 V (n) 1,m 2 .
(6.160)

In fact, three matrices

W (n) 1,m , S (n)
1,m and V k+1,m can be expressed by

S (n) k+1,m = λS (n) τ,m + i p (n) k+1,m (i)w ⊤ i w i , (6.161)
where w i is the i-th row of W

(n) k+1,m . Thanks to Woodbury matrix identity [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] and S

(n) k+1,m ∥ F ≤ √ r/δ < ∞, i.e., V (n)
k+1,m is bounded. As a result, u

k+1,m is bounded for all m = 1, 2, . . . , I n . Thanks to the mathematical induction, we can conclude that the solution U (n) t generated by RACP is bounded for t ≥ 1.

Forward Monotonicity: ft (D t-1 ) ≥ ft (D t ).

We have

ft (D t-1 ) -ft (D t ) =            N n=1 ft U (1) t-1 , . . . , U (n-1) t-1 , U (n) t-1 , . . . , U (N ) t-1 -ft U (1) t-1 , . . . , U (n-1) t-1 , U (i) t , . . . , U (N ) t-1 [Jacobi] N n=1 ft U (1) t , . . . , U (n-1) t , U (n) t-1 , . . . , U (N ) t-1 -ft U (1) t , . . . , U (n-1) t , U (n) t , . . . , U (N ) t-1 [Gauss-Seidel] (6.162) Recall that U (n) t is the minimizer of ft U (1) t-1 , . . . , U (n-1) t-1 , U, U (n+1) t-1 , . . . , U (N ) t-1 if using Jacobi scheme or ft U (1) t , . . . , U (n-1) t , U, U (n+1) t-1 , . . . , U (N )
t-1 if using Gauss-Seidel scheme. Therefore, we always have

ft U (1) t-1 , . . . , U (n-1) t-1 , U (n) t-1 , . . . , U (N ) t-1 ≥ ft U (1) t-1 , . . . , U (n-1) t-1 , U (i) t , . . . , U (N ) t-1 [Jacobi] ft U (1) t-1 , . . . , U (n-1) t-1 , U (n) t-1 , . . . , U (N ) t-1 ≥ ft U (1) t-1 , . . . , U (n-1) t-1 , U (i) t , . . . , U (N ) t-1 [Gauss-Seidel]
As a result, ft (D t-1 ) ≥ ft (D t ).

Backward Monotonicity: ft (D t ) ≤ ft (D t+1 ).

Applying the similar argument above, we also obtain ft (D t ) ≤ ft (D t+1 ).

APPENDIX

Stability of Estimates: ∥D

t -D t-1 ∥ F = O(1/t).
We first prove that the surrogate ft (.) w.r.t. each factor is Lipschitz continuous. Since U (n) t = argmin ft (U (n) , .), we have ft (U

(n) t , .) ≤ ft (U (n) t-1 , .)∀t and hence ft-1 U (n) t , . -ft-1 U (n) t-1 , . ≤ ft-1 U (n) t , . -ft U (n) t , . -ft-1 U (n) t-1 , . -ft U (n) t-1 , . . (6.163)
Lets denote the error function d t (U (n) , .) = ft-1 (U (n) , .) -ft (U (n) , .). We have

∇d t U (n) , . = U (n) A t-1 t -1 - A t t + B t-1 t -1 - B t t , (6.164)
where

A t = t τ =1 β t-τ W (n) τ ⊤ W (n) τ , B t = t τ =1 β t-τ P (n) τ ⊛ (Y (n) τ -O (n) τ ) W (n)
τ . Thanks to the two facts that ∥MN∥ F ≤ ∥M∥ F ∥N∥ F and ∥M + N∥ F ≤ ∥M∥ F + ∥N∥ F [START_REF] Golub | Matrix Computations[END_REF], we obtain

∇d t (U (n) , .) F ≤ κ U A t-1 t -1 - A t t F + B t-1 t -1 - B t t F = c n , (6.165)
where κ U is the upper bound for ∥U (n) ∥ F . As a result, the error function

d t (U (n) ) is Lipschitz with parameter c n = O(1/t), i.e., ft-1 U (n) t , . -ft-1 U (n) t-1 , . ≤ d t U (n) t , . -d t U (n) t-1 , . ≤ c n U (n) t -U (n) t-1 F . (6.166)
Moreover, ft (U (n) , .) is a m-strongly convex function, i.e., ft-1

U (n) t , . -ft-1 U (n) t-1 , . ≥ m U (n) t - U (n) t-1 2 F .
From that, we obtain the asymptotic variation of U (n) as follows U

(n) t -U (n) t-1 F ≤ cn m = O 1/t , Therefore, we can conclude that N n=1 U (n) t -U (n) t-1 2 F = ∥D t -D t-1 ∥ 2 F = O 1/t 2 or ∥D t -D t-1 ∥ F = O 1/t .
Stability of Errors: e t (D t ) -e t-1 (D t-1 ) = O(1/t).

We begin with verifying the differentiable property of the loss function ℓ(D, P t , Y t ) at time t. where H = I N -1 n=1 × n U (n) . We obtain that ℓ(D, P t , Y t ) = min u,O l(D, P t , Y t , O, u) is a continuously differentiable function and its partial derivative w.r.t. U (n) is given by

∂ℓ(D, P t , Y t ) ∂U (n) = 2P (n) t ⊛ Y (n) t -O (n) t -U (n) W(n) t ⊤ W(n) t , (6.168)
where Accordingly, the sum

W(n) t = N -1 i=1,i̸ =n U (i) t-1 ⊙ (u * t ) ⊤ . ( 6
f t (D) = 1/L t t τ =t-Lt+1 β t-τ ℓ(D, P τ , Y τ ) is continuously differen- tiable. Let us denote ft U (n) , . = f t-1 U (n) , . -f t U (n) , .
. Applying the same arguments in subsection I.4, we also obtain

∇ ft U (n) , . F ≤ κ U Ā(n) t-1 t -1 - Ā(n) t t F + B(n) t-1 t -1 - B(n) t t F = d n , (6.170)
where

Ā(n) t = t τ =1 β t-τ W(n) τ ⊤ W(n) τ , and
B(n) t = t τ =1 β t-τ P (n) τ ⊛ (Y (n) τ -O (n) τ ) W(n) τ . Accordingly, ∇ ft U (n) , . is bounded and hence f t U (n) t-1 , . -f t U (n) t , . ≤ d n U (n) t-1 -U (n) t F . ( 6
(D t-1 ) = ft (D t ) -f t (D t ) -ft-1 (D t-1 ) -f t-1 (D t-1 ) ≤ ft (D t ) -ft (D t-1 ) + f t (D t ) -f t (D t-1 ) ≤ N n=1 (c n + d n ) U (n) t-1 -U (n) t F = O(1/t).
(6.172)

It ends the proof.

6.6. APPENDIX

Appendix D: Proof of Lemma 12

Detailed Proof: We apply the similar arguments of Proposition 7 in our companion work [29] to prove Lemma 12.

Almost sure convergence of { ft (D t )} ∞ t=1

Main approach: We prove the convergence of the sequence ft (D t ) by showing that the stochastic positive process u t := ft (D t ) is a quasi-martingale Fisk. In particular, if the sum of the positive difference of u t is bounded, u t is a quasi-martingale, and the sum converges almost surely, thanks to the following quasi-martingale theorem:

Proposition 18 (Quasi-martingale Theorem [343, Theorem 9.4 & Proposition 9.5] and [START_REF] Bottou | Online learning and stochastic approximations[END_REF]Section 4.4]). Let (Ω, F, P) be a probability space, {u t } t>0 be a stochastic process on the probability space and {F t } t>0 be a filtration by the past information at time instant t.

Let us define the indicator function δ t as follows Next, let us define the following indicator function

δ t ∆ =    1 if E[u t+1 -u t |F t ] > 0, 0 otherwise. For all t, if u t ≥ 0 and ∞ i=1 E[δ i (u i+1 -u i )|F i ] < ∞,
δ t ∆ =    1 if E ft+1 (D t+1 ) -ft (D t )|F t > 0, 0 otherwise. (6.176)
Here, the process {δ t } t>0 is adapted to the filtration {F t } t>0 as δ t is measurable with respect to F t for very t. From (D4), we then obtain Accordingly, we obtain

E δ t E ft+1 (D t+1 ) -ft (D t )|F t ≤ E √ t f (D t ) -f t (D t ) 1 √ t(t+1)
. We know that the centered and scaled version of We know that ft (D) is a majorant function of f t (D), i.e.,

f t (D t ) satisfies E √ t f (D t )-f t (D t ) = O(
ft (D + a t V) ≥ f t (D + a t V) ∀D, V ∈ D, a t . (6.180)
Taking the Taylor expansion of (6.180) at t → ∞ results in This property is proved by applying immediately the following stages:

f ∞ D + tr a t V ⊤ ∇f ∞ D + o a t V ≤ f∞ D + tr a t V ⊤ ∇ f∞ D + o a t V , ( 6
1. Stage 1: lim t→∞ tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 = 0; 2. Stage 2: tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 ≤ c 1 tr (D -D t ) ⊤ ∇ ft+1 D t +c 2 D t+1 -D t 2 F ∀t, D ∈ D; 3. Stage 3: ∇ ft ( D)
⊤ (D-D) ⪰ 0 ∀D where D is the limited point of the sequence {D t } t≥1 .

6.6. APPENDIX

Stage 1:

When L t = t, we can recast the surrogate function ft (.) into the following form

ft (D) = ρ t tr A t (U (N ) ) ⊤ U (N ) ⊛ (U (N -1) ) ⊤ U (N -1) ⊛ • • • ⊛ (U (1) ) ⊤ U (1) - 2ρ t tr B t U (N ) ⊙ U (N -1) ⊙ • • • ⊙ U (1) ⊤ + R X ,O , (6.183)
where A t = λA t-1 + u t u ⊤ t , and B t is the (N + 1)-unfolding matrix of the tensor

B t = λB t-1 + P t ⊛ (Y t -O t ) × N +1 u ⊤ t , and R X ,O = ρ t t τ =1 ∥P t ⊛ Y t ∥ 2 F + 1 t t τ =1 β t-τ ∥O τ ∥ 1 independent of D.
With respect to each factor U (n) , we can further express ft (D) as follows

ft (D) = ρ t tr U (n) ⊤ U (n) A t,n - 2ρ t tr U (n) ⊤ B t,n + R X ,O . (6.184)
Here, the two matrices A t,n and B t,n are given by

A t,n = A t ⊛ (U (1) ) ⊤ U (1) ⊛ • • • ⊛ (U (n-1) ) ⊤ U (n-1) (6.185) ⊛ (U (n+1) ) ⊤ U (n+1) ⊛ • • • ⊛ (U (1) ) ⊤ U (1) , B t,n = r j=1 B (j) t × 1 U (1) (:, j) × 2 • • • × n-1 U (n-1) (:, j) (6.186)
× n+1 U (n+1) (:, j) • • • × N U (N ) (:, j),
where B

(j)

t ∈ R I 1 ×I
ft+1 D t -ft+1 D t+1 -tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) ≤ L D t -D t+1 2 F , (6.187)
with L = max n ( Lt,n /2). Thanks to the triangle inequality, we then obtain

tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) ≤ L D t -D t+1 2 F + ft+1 (D t ) -ft+1 (D t+1 ). (6.188)
Accordingly, we have

∞ t=1 E tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) F t ≤ L ∞ t=1 E D t -D t+1 2 F + ∞ t=1 E ft+1 (D t+1 ) -ft+1 (D t ) F t . (6.189) Recall that D t -D t+1 ∥ F = O(1/t) as indicated in Proposition 1, hence ∞ t=1 D t -D t+1 ∥ 2 F ≤ d ∞ t=1 1 t 2 = d π 6 < ∞ for some constant d > 0.
Together with (6.177), we obtain that the right 6.6. APPENDIX hand side of (6.189) is finite.

Also, it is well-known that E[|x|] < ∞ implies |x| < ∞ almost surely for any random variable

x, thus we obtain .190) Moreover, we always have .191) Therefore the series tr[(D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 )] t≥1 converges and we suppose that it converges to C < ∞. Now, we rewrite (6.191) as follows

∞ t=1 tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) < ∞. ( 6
∞ t=1 tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) < ∞ t=1 tr (D t -D t+1 ) ⊤ ∇ ft+1 (D t+1 ) < ∞. ( 6
lim t→∞ t τ =1 tr (D k -D k+1 ) ⊤ ∇ fk+1 (D k+1 ) = lim t→∞ tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 + lim t→∞ t-1 τ =1 tr (D k -D k+1 ) ⊤ ∇ fk+1 (D k+1 ) = C < ∞. (6.192)
When t → ∞, the following partial sum also converges to C, i.e., .193) It implies that

lim t→∞ t-1 τ =1 tr (D k -D k+1 ) ⊤ ∇ fk+1 (D k+1 ) = C. ( 6
lim t→∞ tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 = 0. (6.194)
Step 2: .195) Without loss of generality, we suppose that D is arbitrarily chosen in D such that ∥D - .196) Collecting these inequalities with n = 1, 2, . . . , N together, we derive .197) It then follows that

Because U (n) t+1 = argmin U (n) ft+1 U (n) , . , we have ft+1 U (n) t+1 , . ≤ ft+1 U (n) t + d 1 tN U (n) -U (n) t , . ∀D ∈ D. ( 6
D t ∥ F = d 1 /tN for some positive constant d 1 > 0, hence ∥U (n) -U (n) t ∥ F ≤ d 1 /N t ∀n. As mentioned in Stage 1, ∇ f = ∇ 1 f , ∇ 2 f , . . . ,
(n) t -U (n) t+1 ⊤ ∇ n ft+1 U (n) t+1 , . ≤ d 1 tN tr U (n) -U (n) t ⊤ ∇ n ft+1 U (n) t , . + Ld 2 t 2 N 2 . ( 6
tr (D t -D t+1 ) ⊤ ∇ 1 ft+1 U (1) t+1 , . , ∇ 2 ft+1 U (2) t+1 , . , . . . , ∇ N ft+1 U (N ) t+1 , . ≤ d 1 tN tr D -D t ⊤ ∇ 1 ft+1 U (n) t , . , ∇ 2 ft+1 U (n) t , . , . . . , ∇ N ft+1 U (n) t , . + Ld 2 t 2 N 2 . ( 6
tr (D t -D t+1 ) ⊤ ∇ ft+1 D t+1 ≤ d 1 tN tr (D -D t ) ⊤ ∇ ft+1 D t + Ld 2 D t -D t+1 2 F , (6.198)
because of ∥D t -D t+1 ∥ F = O(1/t). The inequality (6.198) still holds for all D ∈ D such that

∥D -D t ∥ F > d 1 /tN .
Step 3:

We use the proof by contradiction to indicate that the limited point D is a stationary point of f∞ (.) over D.

Assume that D is not a stationary point of ft over D when t → ∞. Then there exists D ′ ∈ D and ϵ 1 > 0 such that .199) Thanks to the triangle inequality, we have According to (6.198), we obtain

tr (D ′ -D) ⊤ ∇ f∞ D ≤ -ϵ 1 < 0. ( 6
(D ′ -D τ ) ⊤ ∇ fk+1 (D τ ) -(D ′ -D) ⊤ ∇ f∞ ( D) F ≤ ∇ fk+1 (D τ ) -∇ f∞ ( D) F D ′ -D τ F + ∥ f∞ ( D)∥ F ∥ D -D τ ∥ F . ( 6
lim k→∞ tr (D τ -D k+1 ) ⊤ ∇ fk+1 (D k+1 ) ≤ -d 1 ϵ tN ∥D ′ -D τ ∥ F < 0, (6.202)
which is a contradiction in (6.194) Suppose that ∀u ∈ U, the function f (., u) is differentiable and

a i = ∞ and ∞ i=1 a i b i < ∞, |b t+1 -b t | < Ka t with some constant K, then lim t→∞ b t = 0 or ∞ i=1 b i < ∞.
∇ v f (v, u) is continuous on V × U. If g(v) be the function derived from g(v) = min u∈U f (v, u), then g(v) is also differentiable. In addition, if u * = argmin u∈U f (v, u) be unique, ∇g(v) = ∇ v f (v, u * ), ∀v ∈ V.
f t = 1 t t i=1 ℓ(U i ), and f = E[f t (U)].
Assume that for all ℓ, ∥ℓ∥ ∞ < M and random variables {U i } i≥1 are Borel-measurable, we then have

E[ √ t∥f t -f ∥ ∞ ] = O(1), where ∥ℓ∥ ∞ ∆ = inf{C ≥ 0, |f (x)| < C ∀ x}.
Proposition 27 (Quasi Martingales [START_REF] Bottou | Online learning and stochastic approximations[END_REF]Section 4.4]). Let (Ω, F, P) be a probability space, {u t } t>0 be a stochastic process on the probability space and {F t } t>0 be a filtration by the past information at time instant t. Let us define the indicator function δ t as follows

δ t ∆ =    1 if E[u t+1 -u t |F t ] > 0, 0 otherwise. For all t, if u t ≥ 0 and ∞ i=1 E[δ i (u i+1 -u i )|F i ] < ∞, then u t is a quasi-martingale and converges almost surely, i.e., ∞ t=1 E[u t+1 -u t |F t ] < ∞.

INTRODUCTION

Tensor-train (TT) decomposition has been an efficient tool to find low order approximation of large-scale, high-order tensors. In online setting, TT decomposition has not gained much attention and popularity as CP and Tucker decompositions. In particular, the existing TT decomposition algorithms are either of high computational complexity or operating in batch-mode, and hence, they become inefficient for (near) real-time processing. In this chapter, we introduce three new online algorithms for the problem of streaming tensor-train decomposition. The first algorithm called TT-FOA is capable of tracking the low-rank components of high-order tensors from noisy and high-dimensional data with high accuracy, even when they come from time-dependent observations. The second algorithm called ATT is specifically designed for handling incomplete streaming tensors. ATT is scalable, effective, and adept at estimating low TT-rank component of streaming tensors. To deal with sparse outliers, we propose the so-called ROBOT algorithm which stands for ROBust Online Tensor-Train decomposition. Technically, ROBOT has the ability to tracking streaming tensors from imperfect streams (i.e., due to noise, outliers, and missing data) as well as tracking their time variation in dynamic environments. We conduct several experiments on both synthetic and real data to demonstrate the effectiveness of the proposed algorithms.

Introduction

Tensor decomposition has received increasing attention from the machine learning and signal processing community over the years [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF]. It has been successfully applied to a broad range of applications, from wireless communications [START_REF] Chen | Tensor decompositions in wireless communications and MIMO radar[END_REF][START_REF] Kofidis | A tensor-based approach to joint channel Estimation/Data detection in flexible multicarrier MIMO systems[END_REF] and image processing [START_REF] Miguel | Nonnegative tensor CP decomposition of hyperspectral data[END_REF][START_REF] Velasco-Forero | Classification of hyperspectral images by tensor modeling and additive morphological decomposition[END_REF] to neuroscience [START_REF] Thanh | Multi-channel EEG epileptic spike detection by a new method of tensor decomposition[END_REF][START_REF] Acar | Multiway analysis of epilepsy tensors[END_REF]. Tensor-train (TT) decomposition, which is one form of tensor decomposition, has become a powerful processing tool for multi-dimensional and large-scale data analysis [START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions[END_REF].

Under the tensor-train format, we can factorize a high-order tensor into a sequence of 3-order tensors, see Fig. 7.1 for an illustration.

TT decomposition offers several advantages compared to the two standard Tucker and CP/PARAFAC decompositions. First, we can represent any high-order tensor under TT decomposition and its computation is stable since it is based on computing low-rank approximations of unfolding matrices of the tensor [START_REF] Oseledets | Tensor-train decomposition[END_REF]. Second, TT-rank can be effectively determined in a stable way in contrast to CP-rank which is known as an NP-hard problem [START_REF] Hillar | Most tensor problems are NP-hard[END_REF][START_REF] Hastad | Tensor rank is NP-complete[END_REF]. Moreover, TT decomposition provides a memory-saving representation for high-order tensors and can break the curse of dimensionality which limits the order of the tensors to be analysed [START_REF] Oseledets | Tensor-train decomposition[END_REF][START_REF] Vervliet | Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis[END_REF]. Accordingly, TT decomposition is expected to be capable of handling big tensors efficiently and effectively.

We refer the readers to [START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions[END_REF] for a comprehensive survey on basic properties, algorithms, and applications of the tensor-train decomposition.

In recent years, the demand for big data stream analysis has been increasing rapidly [2].

7.1. INTRODUCTION  1 1  1 2  1 N  (1) G (2) G ( 1) N  G 1 1 N    11 [] Ir  1 2 2 [ ] r I r  2 1 1 [ ] N N N r I r     1 [] N N rI   () N G X Figure 7.1: Tensor-train decomposition of X ∈ R I 1 ×I 2 ו••×I N .
In most modern online applications, data acquisition is a time-varying process where data are sequentially acquired at a large scale with many attributes over time. This leads to several issues for tensor decomposition in general and TT decomposition in particular: (i) size of the tensor is growing linearly with time, (ii) time variation in nonstationary environments where the underlying process generating the tensor can change over time, and (iii) uncertainties (e.g., imprecise, noisy, and misleading entries) emanate during data collection, to name a few. In parallel, missing data are ubiquitous in multi-dimensional and large-scale data analysis where collecting all data attributes at a time is either too expensive or even impossible due to corruption [START_REF] Little | Statistical Analysis with Missing Data[END_REF].

Accordingly, it is of great interest to develop adaptive (online) tensor decomposition or tensor tracking algorithms which are capable of handling these issues. In spite of several successes in appends the estimated cores to old estimates from past observations. In [START_REF] Wang | ADTT: A highly efficient distributed tensor-train decomposition method for IIoT big data[END_REF], Wang et al. also developed an incremental TT method for factorizing tensors derived from industrial IoT data streams, namely AITT. By exploiting a relationship between the directly reshaped matrix and integration of unfolding matrices, AITT can estimate effectively the underlying TT-cores with low cost. Nevertheless, it is worth noting that the framework of both iTTD and AITT is not really online streaming learning, but incremental batch learning. These drawbacks encourage us to develop adaptive methods for factorizing high-order streaming tensors under the tensor-train format.

STREAMING TENSOR-TRAIN DECOMPOSITION

 1 1  1 2  1 N  (1) t G (2) t G ( 1) N t  G () 1 N t  G 1 1 N      1 t  X t Y 1 1 [ ] Ir  1 2 2 [ ] r I r  2 1 1 [ ] N N N r I r      1 [ ] t N N r I   () N t G t X Figure 7.2: Streaming Tensor-Train Decomposition of X t ∈ R I 1 ×I 2 ו••×I N -1 ×I t N .
7.2 Streaming Tensor-Train Decomposition

Problem Formulation

Consider a streaming N -order tensor X t ∈ R I 1 ×I 2 ו••×I N -1 ×I t N fixing all but the last "time" dimension I t N . At time t, X t is particularly obtained by appending a new slice Y t ∈ R I 1 ×I 2 ו••×I N -1 to the previous observation X t-1 along the time dimension, i.e., I t N = I t-1 N +1, please Fig. 7.2 for an illustration. Instead of recomputing the batch TT decomposition for X t , we aim to develop an efficient update, both in computational complexity and memory storage, to obtain TT-cores of X t from past estimations.

TT decomposition of X t can be represented by a multilinear product of 3-order tensors called TT-cores:

X t = G (1) t × 1 2 G (2) t × 1 3 • • • × 1 N G (N ) t , (7.1)
where r TT = [r 1 , r 2 , . . . , r N -1 ] is a vector containing the TT-ranks, G

∈ R I 1 ×r 1 , G (N ) t ∈ R r N -1 ×I t N and G (n) t ∈ R r n-1 ×In×rn , n = 2, . (1) t
. . , N -1, are the TT-cores. In practice, (7.1) is only an approximate model in a noisy environment, i.e.,

X t = G (1) t × 1 2 G (2) t × 1 3 • • • × 1 N G (N ) t + N t (7.2)
where N t is a noise tensor. The TT-cores can be estimated by solving the following minimization:

G (n) t N n=1 = argmin {G (n) } N n=1 1 2 X t -X 2 F s.t. X = G (1) × 1 2 G (2) × 1 3 • • • × 1 N G (N ) . (7.3) Problem (7.
3) can be rewritten in the adaptive scheme as follows

G (n) t N n=1 = argmin {G (n) } N n=1 t τ =1 β t-τ Y τ -G (1) × 1 2 • • • × 1 N -1 G (N -1) × 1 N g (N ) τ 2 F , (7.4
Input: Observations {Y t } ∞ t=1 , Y t ∈ R I1×I2ו••×I N -1 , TT-rank r TT = [r 1 , r 2 , . . . , r N -1 ], forgetting factor 0 < β ≤ 1. Initialization: {G (n) 0 } N -1
n=1 are initialized randomly and {S

(n) 0 } N -1 n=1 = I. Main Program: Procedure: for t = 1, 2, . . . do Step 1: Estimate g (N ) t H t-1 = G (1) t-1 × 1 2 G (2) t-1 × 1 3 • • • × 1 N -1 G (N -1) t-1 H t-1 = unfolding(H t-1 , [I 1 I 2 . . . I N -1 , r N -1 ]) Ω = randsample [1, I 1 I 2 . . . I N -1 ] y Ωt = vec(Y t ) g (N ) t = H # Ωt-1 y Ωt ∆ t = Y t -H t-1 × 1 N g (N ) t Step 2: Update TT-cores G k in parallel A (n) t-1 = G (1) t-1 × 1 2 • • • × 1 n-1 G (n-1) t-1 A (n) t-1 = unfolding A (n) t-1 , [r n-1 , I 1 I 2 . . . I n-1 ] B (n) t = G (n+1) t-1 × 1 n+2 • • • × 1 N -1 G (N -1) t-1 × 1 N g (N ) t B (n) t = unfolding B (n) t , [r n , I n+1 I n+2 . . . I N -1 ] W (n) t = B (n) t ⊗ A (n) t-1 S (n) t = βS (n) t-1 + W (n) t W (n) t ⊤ V (n) t = S (n) t -1 W (n) t ⊤ ∆ (n) t = unfolding(∆ t , [I n , r n-1 r n ]) G (n) t = G (n) t-1 + ∆ (n) t V (n) t ⊤ G (n) t = reshape(G (n) t , [r n-1 , I n , r n ]) end Output: TT-cores {G (n) t } N n=1 .
TT-cores; (ii) then we update TT-cores G (n) in parallel, given g (N ) t and remaining TT-cores.

The pseudocode of TT-FOA is summarized in Algorithm 9.

Estimation of g (N ) t

Given a new slice Y t and past estimated TT-cores, g

(N ) t can be estimated by solving (7.7)

g (N ) t = argmin g (N ) ∈R r n-1 ×1 Y t -H t-1 × 1 N g (N ) 2 F + ρ 2 g (N ) 2 2 ,
where ρ is a small positive parameter for regularization. It can be reformulated via its matrixvector representation as follows

g (N ) t = argmin g (N ) ∈R r N -1 ×1 y t -H t-1 g (N ) 2 2 + ρ 2 g (N ) 2 2 , (7.11)
where y t = vec(Y t ) and H t-1 ∈ R I 1 ...I N -1 ×r N -1 is the unfolding matrix of H t-1 .

Problem (7.11) is an overdetermined least-squares (LS) regression, it can be efficiently solved by using the randomized sketching technique [START_REF] Mahoney | Randomized algorithms for matrices and data[END_REF], as can be efficiently updated by applying the ridge regression method to (7.12), whose closed-form is given by g

g (N ) t = argmin g (N ) ∈R r N -1 ×1 L H t-1 g (N ) -L y t 2 2 + ρ 2 g (N ) 2 2 , ( 7
(N ) t = H ⊤ Ω t-1 H Ω t-1 + ρI r N -1 -1 H ⊤ Ω t-1 y Ωt . (7.13)
As a result, the last TT-core G (N ) t is updated as follows

G (N ) t = G (N ) t-1 g (N ) t . ( 7
G (n) t = argmin G (n) ∈R In×rnr n-1 f (G (n) ) = t τ =1 β t-τ Y (n) τ -G (n) W (n) τ 2 F , (7.15)
where

G (n) t is the mode-2 matricization of G (n) t , Y (n) τ is the mode-n matricization of Y τ ; W (n) τ = B (n) τ ⊗ A (n)
t-1 where ⊗ denotes the Kronecker product, A The local optimal G (n) t can be obtained by setting the first derivative of f (G (n) ) to zero:

G (n) t i=1 β t-τ W (n) τ W (n) τ ⊤ = t i=1 β t-τ Y (n) τ W (n) τ ⊤ . ( 7
(n) t = t τ =1 β t-τ W (n) τ W (n) τ ⊤ and R (n) t = t τ =1 β t-τ Y (n) τ W (n) τ ⊤ . The two ma- trices R (n) t
and S

(n) t can be updated recursively:

S (n) t = βS (n) t-1 + W (n) t W (n) t ⊤ , (7.17)
R (n) t = βR (n) t-1 + X(k) t W (n) t ⊤ . (7.18)
Therefore, (7.16) can be rewritten as

G (n) S (n) t = βR (n) t-1 + Y (n) t W (n) t ⊤ = βG (n) t-1 S (n) t-1 + Y (n) t W (n) t ⊤ = G (n) t-1 S (n) t + Y (n) t -G (n) t-1 W (n) t W (n) t ⊤ . (7.19)
Let the residual matrix ∆

(n) t and coefficient matrix V (n) t be ∆ (n) t = Y (n) t -G (n) t-1 W (n) t , (7.20)
V (n) t = W (n) t ⊤ S (n) t -1 . (7.21)
We obtain a simple rule for updating G

(n) t as follows G (n) t = G (n) t-1 + ∆ (n) t V (n) t . (7.22)
After that, the TT-core G

(n) t will be derived from reshaping G

(n) t into a 3-way tensor of size

r n-1 × I n × r n .
We also note that when dealing with large-scale and high-rank tensors (i.e. r n ≈ I n ), TT-FOA can be sped up by using its stochastic approximation. We refer to this method as the stochastic TT-FOA. Particularly, the gradient ∇f (G (n) ) can be approximated by the instantaneous gradient of the last summand of f (G (n) ). Thus, S

(n) t can be computed by

S (n) t ≃ W (n) t (W (n) t ) ⊤ . ( 7
Input: Streams {P t ⊛ Y t } ∞ t=1 , P t , Y t ∈ R I1×I2ו••×I N -1 ×W , TT-rank r TT = [r 1 , r 2 , . . . , r N -1 ], forgetting factor 0 < β ≤ 1, regularized parameters ρ, λ > 0. Initialization: {G (n) 0 } N -1 n=1 are initialized at random, {S (n) 0 } N -1 n=1 = 0 and {∆G (n) 0 } N -1 n=1 = 0. Main Program: Procedure: for t = 1, 2, . . . do Stage 1: Estimate the temporal TT-core G (N ) t H t-1 = G (1) t-1 × 1 2 • • • × 1 N -1 G (N -1) t-1 H t-1 = reshape H t-1 , [I 1 I 2 . . . I N -1 , r N -1 ] for i = 1, 2, . . . , W do y t,i = vec Y t (:, . . . , :, i) Pt,i = diag P t (:, . . . , :, i) G (N ) t (:, i) = H ⊤ t-1 Pt,i H t-1 + λI r N -1 -1 H ⊤ t-1 Pt,i y t,i δy t,i = Pt,i y t,i -H t-1 G (N ) t (:, i) ∆Y t,i = reshape δy t,i , [I 1 , I 2 , . . . , I N -1 , 1] end G (N ) t = G (N ) t-1 G (N ) t ∆Y t = ∆Y t,1 ⊞ N ∆Y t,2 ⊞ N • • • ⊞ N ∆Y t,W Stage 2: Estimate the non-temporal TT-cores G (n) t N -1 n=1 for n = 1, 2, . . . , N -1 do A (n) t-1 = G (1) t-1 × 1 2 • • • × 1 n-1 G (n-1) t-1 A (n) t-1 = reshape A (n) t-1 , [r n-1 , I 1 I 2 . . . I n-1 ] B (n) t = G (n+1) t-1 × 1 n+2 . . . G (N -1) t-1 × 1 N G (N ) t B (n) t = reshape B (n) t , [r n , I n+1 I n+2 . . . I N -1 ] W (n) t = B (n) t ⊗ A (n) t-1 S (n) t = βS (n) t-1 + W (n) t (W (n) t ) ⊤ ∆G (n) t = P (n) t ⊛ ∆Y (n) t W (n) t ⊤ + βρ∆G (n) t-1 S (n) t + ρI rn-1rn -⊤ G (n) t = G (n) t-1 + ∆G (n) t G (n) t = reshape G (n) t , [r n-1 , I n , r n ] end Stage 3 (Optional): Re-estimate G (N ) t with updated G (n) t N -1 n=1 as in Stage 1. end Output: TT-cores G (n) t N n=1 . estimations G (n) t-1 N -1 n=1 ; and second, estimate the non-temporal G (n) t given G (N ) t
and remaining TT-cores, for n = 1, 2, . . . , N -1. In stage 1, we apply the well-known regularized least-squares method for estimating G (N )

t . An elegant recursive least-squares (RLS) adaptive filter is specifically developed to update the non-temporal TT-cores {G

(n) t } N -1
n=1 in an effective way. Main steps of the proposed ATT method are summarized in Algorithm 10. On the arrival of Y t , we obtain G

(N ) t from G (N ) t = argmin G (N ) P t ⊛ Y t -H t-1 × 1 N G (N ) 2 F + λ G (N ) 2 F , (7.28)
where

H t-1 = G (1) t-1 × 1 2 • • •× 1 N -1 G (N -1) t-1
and λ > 0 is a small regularized parameter. Here, the first term of (7.28) is aimed at minimizing the residual error between observation and estimation for t-th temporal slice, while the introduction of λ∥G (N ) ∥ 2 F is for avoiding the ill-posed computation in practice. Particularly, we can rewrite (7.28) as follows

G (N ) t = argmin G (N ) P t ⊛ Y t -H t-1 G (N ) 2 2 + λ G (N ) 2 F , (7.29)
where Y t , P t ∈ R I 1 ...I N -1 ×W , and H t-1 ∈ R I 1 .. We update

{G (n) } N -1 n=1 by minimizing G (n) t = argmin G (n) t τ =1 β t-τ P τ ⊛ Y τ -A (n) t-1 × 1 n G (n) × 1 n+1 B (n) τ 2 F + ρ G (n) -G (n) t-1 2 F , (7.32)
where

A (n) t-1 = G (1) t-1 × 1 2 • • • × 1 n-1 G (n-1) t-1 and B (n) τ = G (n+1) t-1 × 1 n+2 • • • × 1 N -1 G (N -1) t-1 × 1 N G (N )
τ . For a better interpretation, we further recast (7.32) as

G (n) t = argmin G (n) t τ =1 β t-τ P (n) τ ⊛ Y (n) τ -G (n) W (n) τ 2 F + ρ G (n) -G (n) t-1 2 F , (7.33)
where

G (n) t = reshape G (n) t , [I n , r n-1 r n ] ; P (n) τ , Y (n) τ
are the mode-n unfolding matrices of P τ and Y τ ; W

(n) τ = B (n) τ ⊗ A (n) t-1 where A (n) t-1 = reshape A (n) t-1 , [r n-1 , I 1 I 2 . . . I n-1 ] (7.34) B (n) τ = reshape B (n) t , [r n , I n+1 I n+2 . . . I N -1 ] (7.35)
Similar to the update of G (N ) t in the first stage, we can update independently each row g

(n) t,m of G (n) t
as follows:

g (n) t,m = argmin g (n) m t τ =1 β t-τ P(n) τ,m y (n) τ,m -g (n) m W (n) τ ⊤ 2 2 + ρ g (n) m -g (n) t-1,m 2 2 , (7.36)
where y

(n) τ,m = Y (n) τ (m, :) and P(n) τ,m = diag P (n) τ (m, :) . Specifically, g (n)
t,m can be derived from setting the gradient of the function in (7.36) to zero:

ρI r n-1 rn + t τ =1 β t-τ W (n) τ P(n) τ,m W (n) τ ⊤ g (n) m ⊤ = ρ g (n) t-1,m ⊤ + t τ =1 β t-τ W (n) τ P(n) τ,m y (n) τ,m ⊤ . (7.37)
The closed-form solution of (7.37) is then given by g

(n) t,m = S (n) t,m + ρI r n-1 rn -1 d (n) t,m + ρ g (n) t-1,m ⊤ ⊤ , (7.38)
where S

(n)

t,m and d

(n)

t,m can be recursively updated as

S (n) t,m = βS (n) t-1,m + W (n) t P(n) t,m W (n) t ⊤ and d (n) t,m = βd (n) t-1,m + W (n) t P(n) t,m y (n) t,m
⊤ . After doing some simple calculations, we can rewrite (7.38) as

g (n) t,m = g (n) t-1,m + δy (n) t,m P(n) t,m W (n) t ⊤ + βρδg (n) t-1,m S (n) t,m + ρI r n-1 rn -⊤ , (7.39)
where δy

(n) t,m = P(n) t,m y (n) t,m -g (n) t-1,m W (n) t ⊤ and δg (n) t-1,m = g (n) t-1,m -g (n)
t-2,m . Accordingly, a recursive rule with a lower space complexity for updating the whole matrix G (n) t at the same time can be given by

G (n) t = G (n) t-1 + P (n) t ⊛ ∆Y (n) t W (n) t ⊤ + ρ∆G (n) t-1 S (n) t + ρI r n-1 rn -⊤ , (7.40)
where ∆Y

(n) t,m = Y (n) t -G (n) t-1 W (n) t and ∆G (n) t-1 = G (n) t-1 -G (n) t-2 .
Then, we simply set G

(n) t N -1 n=1 , ∆G (n) t N -1 n=1 , and S (n) t N -1 n=1 .
Compared to batch TT methods (e.g., TT-SVD [START_REF] Oseledets | Tensor-train decomposition[END_REF] and TT-HSVD [START_REF] Zniyed | A TT-Based hierarchical framework for decomposing high-order tensors[END_REF]), the cost of ATT is much cheaper as it is independent of the temporal dimension. Besides, its computation involves only cheap matrix-matrix products and inverse operations of small matrices, and hence, it avoids the expensive computation of SVD on the tensor's unfolding matrices. Compared to TT-FOA that is the first and only adaptive algorithm for streaming TT decomposition in the literature, ATT shares the same computational and space complexity.

Streaming Tensor-Train Decomposition with Sparse Outliers

In this paper, we introduce a new tensor-train method for factorizing incomplete high-order streaming tensors possibly corrupted by sparse outliers. The proposed method is referred to as ROBOT which stands for ROBust Online Tensor-Train decomposition. ROBOT involves two well-known optimization methods: block-coordinate descent (BCD) and recursive least-squares (RLS). Thanks to the BCD framework, ROBOT decomposes the main optimization into two stages: (i) online outlier rejection and (ii) tracking of TT-cores in time. In the former stage, we apply an effective ADMM solver to estimate the last (temporal) TT-core and sparse outliers living in observations. In the latter stage, we present an efficient RLS solver to minimize an exponential weighted least-squares objective function accounting for missing entries and time variations of TT-cores. Technically, ROBOT is capable of estimating the low-rank components of the underlying tensor from imperfect streams (i.e., due to noise, outliers, and missing data) and tracking their time variation in dynamic environments. To the best of our knowledge, ROBOT is the first streaming TT decomposition robust to sparse outliers, missing data, and time variation.

Problem Formulation

In this paper, we study the robust adaptive tensor-train decomposition of a N -order streaming tensor X t in the presence of both sparse outliers and missing data. Without loss of generality, we suppose the last dimension of X t is temporal, while the others remain constant with time, i.e.,

X t ∈ R I 1 ×I 2 ו••×I N -1 ×I t N
. Specifically, at time t, X t is obtained by concatenating the incoming data stream

Y t ∈ R I 1 ×I 2 ו••×I N -1 ×W (with W ≥ 1)
to the old observation X t-1 along the temporal dimension I t N , i.e.,

X t = X t-1 ⊞ N Y t and I t N = I t-1 N + W. (7.41)
The temporal slice Y t is supposed to have the form

Y t = P t ⊛ L t + O t + N t , (7.42
L t = G (1) t × 1 2 G (2) t × 1 3 • • • × 1 N G (N ) t , (7.43)
where G

(n) t ∈ R r n-1 ×In×rn for n = 1, 2, . . . , N with r 0 = r N = 1 is the n-th TT-core; [r 1 , r 2 , . . . , r N -1 ] is called TT-rank; and G

(N ) t ∈ R r N -1 ×W contains the last W columns of G (N ) t .
In online settings, we propose to minimize the following objective function:

argmin {G (n) } N n=1 ,O t k=1 β t-k P k ⊛ G (1) × 1 2 • • • × 1 N -1 G (N -1) × 1 N G (n) τ + O k -Y k 2 F + ρ 1 O k 1 + ρ 2 N -1 n=1 G (n) -G (n) t-1 2 F . (7.44)
Here, β ∈ (0, 1] plays the role of a forgetting factor in adaptive filter theory which aims to reduce the impact of distant observations as well as deal with nonstationary environments [START_REF] Haykin | Adaptive Filter Theory[END_REF]. The ℓ 1 -norm enforces the sparsity on O (the outliers), while the last regularization term of (7.44) is to control the time variation of TT-cores between two consecutive instances. In addition, we make two mild assumptions on the data model to support our algorithm development in Section III: TT-cores {G (n) } N -1 n=1 may either be static or vary slowly with time, i.e., G

n) t ≃ G (n) (
t-1 ; and the TT-rank is supposed to be known.

Proposed Method

In this section, we propose an adaptive method called ROBOT (which stands for ROBust Online Tensor-Train) for factorizing tensors derived from data streams in the presence of sparse outliers and missing data. Particularly, we decompose the main problem (7.44) into two stages:

• Stage 1: update G (N ) t and O t given G (n) t-1 N -1 n=1 ; • Stage 2: estimate G (n) t given G (N ) t , O t ,
G (N ) t , O t = argmin G (N ) ,O P t ⊛ H t-1 × 1 N G (N ) + O -Y t 2 F + ρ 1 O 1 + ρ 2 G (N ) 2 F , (7.45)
where

H t-1 = G (1) t-1 × 1 2 • • • × 1 N -1 G (N -1) t-1
and the term ρ 2 ∥G (N ) ∥ 2 F is to mitigate ill matrix conditions. Interestingly, we exploit the fact that (7.45) can be decomposed into W sub-problems w.r.t. W columns of G (N ) t , as follows:

argmin g i ,o i P t,i H t-1 g i + o i -y t,i 2 2 + ρ 1 o i 1 + ρ 2 g i 2 2 .
(7.46)

Here, g i , o i , and y t,i are, respectively, the i-th column of G (N ) , the two unfolding matrices of O and Y t ; the mask P t,i = diag P (N ) t (i, :) ; while the matrix H t-1 ∈ R I 1 ...I N -1 ×r N -1 is a matricization of H t-1 .

Since both ℓ 1 -norm and ℓ 2 -norm are convex, (7.46) can be effectively minimized by several methods, e.g., block coordinate descent (BCD) [START_REF] Xu | A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion[END_REF] and alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. In this work, we adopt the ADMM solver introduced in our companion work on robust subspace tracking [25]. Specifically, the update rule at the j-th iteration of the 7.4. STREAMING TENSOR-TRAIN DECOMPOSITION WITH SPARSE OUTLIERS solver is given by

g j = H ⊤ t-1 P t,i H t-1 + ρ 2 I r N -1 -1 H ⊤ t-1 P t,i y t,i -o j-1 + e j-1 , z j = P t,i H t-1 g j + s j-1 -y t,i , e j = λ 1 1 + λ 1 z j + 1 1 + λ 1 S 1+ 1 λ 1 z j , u j = 1 1 + λ 2 P t,i y t,i -H t-1 g j -λ 2 (o j-1 -r j-1 ), o j = S ρ 1 /λ 2 u j + r j-1 , r j = r j-1 + u j -s j .
Here, z j , e j , u j , r j are dummy variables aiming to accelerate the update initialized as zeros;

the augmented Lagrangian parameters λ 1 and λ 2 can be chosen in the range [1, 1.8]; and S α (.)

is the soft-thresholding operator defined as S α (x) = max(0, x -α) -max(0, -x -α). We refer the readers to [25] for further details. Note that since (7.46) is a biconvex minimization problem, and thus, we can apply any other existing proved algorithm to obtain its optimal solution [START_REF] Gorski | Biconvex sets and optimization with biconvex functions: A survey and extensions[END_REF].

The temporal TT-core G

(N ) t is simply obtained by G (N ) t = [G (N ) t-1 G (N ) t ].
In addition, we can re-update G (N ) t in the same way as above when others TT-cores {G

(n) t } N -1
n=1 are updated. Furthermore, after obtaining the outlier O t , we can accelerate the tracking ability of ROBOT by re-updating the observation mask P t as follows

P t i 1 i 2 ...i N =    0, if O t i 1 i 2 ...i N ̸ = 0, P t i 1 i 2 ...i N , otherwise. (7.47)
It is motivated by the following observation: In the literature of robust subspace tracking (RST), the outlier rejection step can facilitate the tracking ability of RST estimators because only "clean" data are involved in the tracking process [25]. Our stage 2 for tracking the TT-cores can be viewed as an extended version of RST for high-order streaming tensors, so the outlier rejection mechanism of (7.47) can improve its performance.

Estimation of TT-cores

G (n) t N -1 n=1 We estimate {G (n) } N -1 n=1 by minimizing G (n) t = argmin G (n) t τ =1 β t-τ P τ ⊛ A (n) t-1 × 1 n G (n) × 1 n+1 B (n) τ -Y τ 2 F + ρ 2 G (n) -G (n) t-1 2 F , (7.48)
where . Accordingly, we can apply the effective recursive least-squares (RLS) method as proposed in our work [30] for minimizing (7.48). For the sake of completeness, we describe here the main steps of the RLS solver and refer the readers to [30] for further details.

A (n) t-1 = G (1) t-1 × 1 2 • • • × 1 n-1 G (n-1) t-1 and B (n) τ = G (n+1) t-1 × 1 n+2 • • • × 1 N -1 G (N -1) t-1 × 1 N G (
For a better interpretation, we first recast (7.48) as

G (n) t = argmin G (n) In m=1 t τ =1 β t-τ P(n) τ,m g (n) m B (n) τ ⊗ A (n) t-1 -y (n) τ,m 2 2 + ρ 2 g (n) m -g (n) t-1,m 2 2 , (7.49)
where g

(n) m is the m-th row of G (n) ∈ R In×r n-1 rn which is the transpose of the mode-2 unfolding matrix of G (n) , Pτ,m = diag P(n) τ (m, :) , A (n) t-1 = reshape A (n) t-1 , [r n-1 , I 1 I 2 . . . I n-1 ] , and
B (n) τ = reshape B (n) t , [r n , I n+1 I n+2 . . . I N -1 ] . Let us denote W (n) τ = B (n) τ ⊗ A (n) t-1 and S (n) τ,m = t τ =1 β t-τ W (n) t P(n) t,m W (n) t ⊤ , (7.50)
d (n) t,m = t τ =1 β t-τ W (n) τ P(n) τ,m y (n) τ,m ⊤ . (7.51)
At time t, we then have

S (n) t,m = βS (n) t-1,m + W (n) t P(n) t,m W (n) t ⊤ (7.52) d (n) t,m , = βd (n) t-1,m + W (n) t P(n) t,m y (n) t,m ⊤ . (7.53)
Setting the gradient of (7.49) to zero results in:

In m=1 S (n) t,m + ρ 2 I r n-1 rn g (n) m ⊤ = In m=1 d (n) t,m + ρ 2 g (n) t-1,m ⊤ . (7.54)
Therefore, we can express each row g

(n) t,m of G (n) t separately as S (n) t,m + ρ 2 I r n-1 rn g (n) t,m ⊤ = d (n) t,m + ρ 2 g (n) t-1,m ⊤ . ( 7
(n) t,m = g (n) t-1,m + δy (n) t,m P(n) t,m W (n) t ⊤ + βρ 2 δg (n) t-1,m S (n) t,m + ρ 2 I r n-1 rn -⊤ , (7.56)
where δy

(n) t,m = P(n) t,m y (n) t,m -g (n) t-1,m W (n) t ⊤ and δg (n) t-1,m = g (n) t-1,m -g (n) t-2,m . Collecting all rows g (n)
t,m together (for m = 1, 2, . . . , I n ), we obtain a simpler recursive rule as

G (n) t = G (n) t-1 + P (n) t ⊛ ∆X (n) t W (n) t ⊤ + βρ 2 ∆G (n) t-1 S (n) t + ρ 2 I r n-1 rn -⊤ , (7.57)
where

∆X (n) t,m = X (n) t -G (n) t-1 W (n) t and ∆G (n) t-1 = G (n) t-1 -G (n) t-2 , and
S (n) t = βS (n) t-1 +W (n) t W (n) t ⊤ .
To enable the recursive update (7.57), we set ∆G

(n) 0 = 0 and S (n) 0 = δ (n) I r n-1 rn with δ (n) > 0.

Computational Complexity and Memory Storage

For short, we suppose I n = I and r n = r for all n = 1, 2, . .

N -1 TT-cores {G (n) t } N -1 n=1 . Therefore, the overall complexity of ROBOT is O r 2 max (N - 1)I N -1 r 2 , W |Ω t | flops. With respect to memory storage, ROBOT requires O (N -1)(2Ir 2 + r 4 ) words of memory for storing G (n) t N -1 n=1 , ∆G (n) t N -1 n=1 , and S (n) t N -1 n=1 .

Experiments

In this section, we conduct several experiments on both synthetic and real data to evaluate the performance of TT-FOA, ATT, and ROBOT for adaptive TT decomposition. Experiments are implemented in MATLAB platform and are available online to facilitate replicability and reproducibility. 1

Performance of TT-FOA

We investigate the tracking ability of TT-FOA with respect to the following aspects: effect of the forgetting factor λ, effect of the noise level σ, its performance in time-varying environments, and its use for real data.

Synthetic Data

We generate streaming 4-way tensors X t ∈ R I1 ×I 2 ×I 3 ×I t 4 of a TT-rank vector r TT = [r 1 , r 2 , r 3 ] as follows:

Y t = G (1) t × 1 2 G (2) t × 1 3 G (3) t × 1 4 g (4)
t + ϵN t , The first task is to track surveillance video. We compare TT-FOA against the two state-ofthe-art adaptive CP tensor decompositions, including PARAFAC-SDT [START_REF] Nion | Adaptive algorithms to track the PARAFAC decomposition of a third-order tensor[END_REF] and OLCP [START_REF] Zhou | Accelerating online CP decompositions for higher order tensors[END_REF].

In order to apply these algorithms effectively, color video frames are converted into grayscale.

Performance of ATT

We investigate the tracking ability of ATT with respect to the following aspects: additive noise effect, and its performance in nonstationary environments. Its effectiveness for real data is We use the following relative error (RE) metric to evaluate the estimation accuracy:

RE Y tr , Y es = Y tr -Y es F Y tr F , (7.60)
where Y tr (resp. Y es ) refers to the true tensor (resp. reconstructed tensor).

Effect of the noise level σ n

In this task, we vary the value of σ n and evaluate the performance of ATT. Here, we used a static tensor (i.e., ε = 0) of size 20 × 20 × 20 × 1000 and rank r TT = [5,5,5]. The missing density ω miss was set to 10%. We fixed the forgetting factor β and the two regularized parameters ρ, λ at 0.5, 1, and 1, respectively. A significant change was also created at t = 600 (i.e., we set ϵ = 1 when t = 600 and ε = 0 otherwise) to investigate how fast ATT could converge. The result is illustrated in Fig. 7.11. We can see that the noise level σ n does not affect the convergence rate of ATT but only its estimation error.

Effect of the time-varying factor ε

We next investigate the tracking ability of ATT in nonstationary environments. Similar to the previous experiment, we also vary the value of ε and then evaluate its estimation accuracy. Most of experimental parameters were kept as above, except the noise level σ n which was set to 10 -3 . Fig. 7.12 illustrates the performance of ATT versus the value of ε. We can see that the estimation accuracy of ATT goes down when ε increases, but converges towards a steady-state error in the similar manner as in the previous case. Intuitively, the time-varying factor has an influence on the convergence rate of tracking algorithms. However, as shown in Fig. 7.12, the value of ε does not affect ATT's convergence rate. This "phenomenon" thus deserves further investigations.

Effect of the missing density ω miss

Here, we measure the performance of ATT in the presence of different missing densities. Particularly, the value of ω miss was chosen among {20%, 40%, 80%}. We reused the same 4-order streaming tensor above with σ n = ε = 10 -3 . Fig. 7.13 shows that the number of missing entries in X t has an impact on both convergence rate and estimation accuracy of ATT, i.e., the lower the value of ω miss is, the better performance ATT achieves. However, even with 80% of missing data, ATT is still able to achieve relatively good performance.

Online video completion

Three real video sequences are used in this task, including "Lobby", "Highway", and "Hall".

Their sizes are summarized in Table 7.1.

We compare ATT with other online tensor completion algorithms: TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], ACP [29],

and ATD [29]. To have a fair comparison, colour video frames were converted into grayscale ones.

The CP-rank, Tucker-rank, and TT-rank were set to 10, [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF], and [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF], respectively.

The results in Table 7.1 (i.e., averaged relative errors) and Fig. 7.14 indicate that ATT provided a competitive video completion performance.

Performance of ROBOT

We here evaluate the performance of ROBOT in terms of the following aspects: (i) impact of noise, (ii) its tracking ability in nonstationary environments, (iii) impact of missing observations, (iv) impact of outliers, and (v) its use for the problem of video background and foreground separation.

Experiment Setup

We follow the problem formulation in Section II to simulate temporal slices {Y t } t≥1 . In particular, Y t is randomly generated under the model

Y t = P t ⊛ L t + O t + N t where L t = G (1)
t × 1 2 G

(2)

t × 1 3 G (3) t × 1 4 g (4)
t .

(7.61)

Here, P t ∈ R I 1 ×I 2 ×I 3 ×1 is a binary mask tensor whose entries are obtained by a Bernoulli model with probability 1 -ω miss (i.e., ω miss represents the missing density). N t is a Gaussian noise 7.5. EXPERIMENTS ∈ R r n-1 ×In×rn and its entries are derived from N (0, 1). At t = 0, G (n) 0 is initialized by a Gaussian distribution with zero mean and unit variance.

To evaluate the performance of ROBOT, we use the following relative error:

RE X tr , X es = X tr -X es F X tr F , (7.62)
where X tr (resp. X es ) refers to the true low-rank component (resp. estimation). set as: time-varying factor ϵ = 0, missing density ω miss = 0%, and outlier density ω outlier = 0% (i.e. outliers free observations). We fixed algorithmic parameters of ROBOT as follows: the forgetting factor β = 0.5 and two penalty parameters ρ 1 = ρ 2 = 1. The result is shown in Fig. 7.15. Clearly, the value of σ n does not affect ROBOT's convergence rate but its relative error.

Effect of the time-varying factor ϵ

Next, we evaluate the performance of ROBOT in dynamic and nonstationary environments. We reused the streaming tensor above with 90% observations (i.e., ω miss = 10%). The noise level σ n was fixed at 10 -3 . We set the outlier density and intensity to 10% and 1, respectively. The forgetting factor and two penalty parameters were kept as above. Also, an abrupt change was made at t = 600 to assess how fast ROBOT converges. Fig. 7.16 illustrates the effect of ϵ on the performance of ROBOT. We can see that the performance of ROBOT increases when ϵ decreases and converges towards a steady-state error.

Effect of the missing density ω miss

We then investigate the tracking ability of ROBOT in the presence of missing data. The value of ω miss was chosen among {10%, 50%, 90%}. We kept all experimental parameters as above, except the time-varying factor ϵ which was set to 10 -3 . We can see from Fig. 7.17 that both convergence rate and estimation accuracy of ROBOT are affected by the value of ω miss . The lower ω miss is, the better performance ROBOT achieves.

Effect of outliers

Here, we measure the robustness of ROBOT against sparse outliers. Most of experimental parameters were kept as in the previous tasks: ω miss = 10%, β = 0.5, σ n = ϵ = 10 -3 , and ρ 1 = ρ 2 = 1. We investigated the case when 30% entries were corrupted by outliers. Three levels of the outlier intensity fac-outlier were considered, including 0.1, 1, and 10 (resp. low, moderate, and strong effect). In this task, 4 we used three video datasets, including "Lobby", "Highway", and "Hall". The dataset "Lobby" includes 1700 frames of size 144 × 176. There are 1700 frames of size 240 × 320 in the data "Highway, while "Hall" consists of 3584 frames whose size is 174 × 144. The performance of ROBOT was evaluated in comparison with two online background/foreground separation algorithms, including PETRELS-ADMM [25] and GRASTA [START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF]. The subspace rank and TTrank were set to 10 and [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF], respectively. The result from Fig. 7.19 indicates that ROBOT is able to detect moving objects in real surveillance video sequences with reasonable performance.

Conclusions

In this chapter, we have considered the problem of tensor tracking under the tensor-train format.

Three novel adaptive tensor-train decomposition algorithms are proposed for factorizing streaming tensors, including TT-FOA, ATT, and ROBOT. Each algorithm is specifically designed for dealing with a specific task. In particular, the former algorithm TT-FOA and its stochastic variant have the capability to track the tensor-train representation of streaming tensors from noisy and high-dimensional data with high accuracy, even when they come from time-dependent observations. By utilizing the recursive least-squares method in adaptive filtering, the second algorithm ATT minimizes effectively a weighted least-squares objective function accounting for both missing values and time-variation constraints on the underlying tensor-train cores. The latter algorithm ROBOT -which is a robust version of ATT -is fully capable of tracking the underlying low-rank component of incomplete streaming tensors corrupted by sparse outliers in

Conclusion and Outlook

Conclusions

In this thesis, we have presented several contributions to the problem of tracking the low-rank approximation of big data streams over time.

For Subspace Tracking

• We provided a survey on recent robust subspace tracking (RLS) algorithms to fill the gap in the literature particularly addressing non-Gaussian noises (i.e., outliers, impulsive noise, and colored noise) and sparse constraints. In the context of missing data and outliers, we reviewed four main classes of RST algorithms, including Grassmannian, recursive leastsquares (RLS), recursive projected compressive sensing (ReProCS), and adaptive projected subgradient method (APSM). When the data streams are corrupted by impulsive noises, we indicated that most of state-of-the-art subspace tracking algorithms are based on improving the well-known PAST algorithm, together with weighted RLS and adaptive Kalman filtering. Next, we outlined two main approaches to deal with subspace tracking in the presence of colored noises, including instrumental variable-based and oblique projections.

Finally, a short review on sparse subspace tracking algorithms was presented.

• We proposed a probable adaptive algorithm called PETRELS-ADMM for tracking the underlying subspace from incomplete observations corrupted by sparse outliers. The proposed algorithm contains two main stages: outlier rejection and subspace estimation. In particular, outliers residing in the measurement data are detected and removed by our ADMM solver in an effective way. Next, we proposed an improved version of PETRELS, namely iPETRELS. It is observed that PETRELS is ineffective when the fraction of missing data is too large. We thus added a regularization of the ℓ 2,∞ -norm, which aims to control the maximum ℓ 2 -norm of rows in the subspace matrix, in the objective function to avoid 8.1. CONCLUSIONS such performance loss. Moreover, we also introduced an adaptive step size to speed up the convergence rate as well as enhance the subspace estimation accuracy. Furthermore, we successfully established a theoretical convergence which guarantees that the solutions generated by PETRELS-ADMM will converge to a stationary point asymptotically.

• We proposed a novel adaptive algorithm called OPIT for the sparse subspace tracking (SST) problem. OPIT takes both advantages of power iteration and thresholding methods, and hence offers several appealing features over the state-of-the-art tracking algorithms. First, OPIT belongs to the class of power methods, and thus its convergence rate is highly competitive compared to other SST algorithms, especially in the high SNR regime. Different from the existing two-stage SST algorithms, OPIT has ability to track the sparse principal subspace with high accuracy in both the classical regime and the HDLSS regime. In addition, OPIT is flexible and very adaptable for different scenarios. For example, we can adjust its procedure for dealing with multiple incoming data streams. Also, it is easy to introduce regularization parameters into OPIT in order to regularize its performance in non-standard environments. Moreover, we can recast its update rule into a column-wise update. Thanks to the deflation transformation, we derived a fast variant of OPIT called OPITd with lower complexity of both computation and memory storage. This variant is fast and useful for tracking high-dimension and large-scale data streams residing in a low-dimensional space. Together with PETRELS-ADMM, OPIT belongs to the class of provable subspace tracking algorithms in which its convergence is guaranteed. Under certain conditions, OPIT can achieve an ϵ-relative-error approximation with high probability when the number of observations is large enough.

For Tensor Tracking

• We provided a comprehensive survey on the state-of-the-art tensor tracking algorithms.

It begins with basic coverage of five common tensor decompositions and their main features, including CP, Tucker, BTD, tensor-train, and t-SVD. Two kinds of streaming models were introduced to represent streaming tensors: single-aspect and multi-aspect. Next, we reviewed four main classes of online CP algorithms: subspace-based, block-coordinate descent, Bayesian inference, and multi-aspect streaming CP decomposition. Under the Tucker format, we categorized the current single-aspect tensor tracking algorithms into two main classes: online tensor dictionary learning and tensor subspace tracking. Multi-aspect streaming Tucker decomposition algorithms were also surveyed. Moreover, we provided a brief survey on other online techniques for tracking tensors under tensor-train, t-SVD, and BTD formats.

• We proposed three efficient adaptive algorithms for tracking the low-rank component of 8.2. RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS streaming tensors over time. Under the CP format, we developed a novel adaptive CP algorithm called ACP for tracking high-order incomplete streaming tensors. ACP is fast and requires a low computational complexity and memory storage, thanks to the alternative minimization and randomized sketching. Under the Tucker format, we proposed the second algorithm, namely adaptive Tucker decomposition (ATD), more flexible than ACP, for the problem of tensor tracking. ATD exhibits competitive performance in terms of both estimation accuracy and computational complexity. Third, we introduced a robust version of ACP called RACP for the problem of tensor tracking in the presence of both missing data and outliers. In particular, RACP aims to learn the low-rank component of streaming tensors in an online fashion as well as offering robustness against gross data corruptions.

More importantly, we proved that ACP, ATD, and RACP are provable algorithms with convergence guarantee.

• We developed three new methods for the problem of tensor tracking under the tensor-train (TT) format. The first method called TT-FOA is capable of tracking the low-rank components of high-order tensors from noisy and high-dimensional data with high accuracy, even when they come from time-dependent observations. The second method called ATT is particularly designed for handling incomplete streaming tensors. ATT is scalable, effective, and adept at estimating low TT-rank component of streaming tensors. Besides, ATT can support parallel and distributed computing. To deal with sparse outliers, we proposed the so-called ROBOT which stands for ROBust Online Tensor-Train decomposition. Technically, ROBOT has the ability to tracking streaming tensors from imperfect streams (i.e., due to noise, outliers, and missing data) as well as tracking their time variation in dynamic environments.

Research Challenges, Open Problems, and Future Directions

In this section, we present several research challenges and open problems that should be considered for the development of tensor tracking problems in the future. These problems also cover the subspace tracking problem as it is a special case of tensor tracking. They are data imperfection and corruption; rank revealing and tracking; efficient and scalable tensor tracking; and other aspects such as theoretical analysis, symbolic data, and tracking under some less common tensor formats. Possible solutions for these challenges are also discussed.

Data Imperfection and Corruption

Dealing with data imperfection and corruption has been a critical issue in many applications and tracking problems in particular [START_REF] Bifet | Adaptive Stream Mining: Pattern Learning and Mining from Evolving Data Streams[END_REF]. We here present two main types of imperfect data that 8.2. RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS either remain unsolved or are still challenging for tensor tracking: (i) non-Gaussian and colored noises; (ii) outliers and missing data.

Non-Gaussian and Colored Noises

Most of the existing tensor tracking algorithms were proposed under the additive white Gaussian noise assumption. This assumption however does not always hold in practice. For example, impulsive noises (e.g., burst, alpha-stable, and spherically invariant random variable noise), which are introduced by human activities and natural sources, are one of the most common non-Gaussian noises that often appear in tracking applications such as direction of arrivals [START_REF] Saucan | CPHD-DOA Tracking of multiple extended sonar targets in impulsive environments[END_REF],

OFDM systems [START_REF] Wang | New results on joint channel and impulsive noise estimation and tracking in underwater acoustic OFDM systems[END_REF] and adaptive system identification [START_REF] Das | Lorentzian based adaptive filters for impulsive noise environments[END_REF]. This type of noise can significantly impact the tracking ability of estimators and it requires specific treatments [26]. In parallel, colored noises that indicate types of noise that are correlated in space and/or time may reduce the performance of tracking algorithms [START_REF] Hänggi | Colored noise in dynamical systems[END_REF]. Accordingly, standard tracking algorithms may be less effective in estimation accuracy in the presence of these noises. They need to be readapted or redesigned for more robustness.

To the best of our knowledge, we are not aware of any tensor tracking algorithm capable of handling such noises in the literature. Some potential approaches have been successfully demonstrated in subspace tracking problems (i.e., tracking tensors of order 2), see [26] for a brief survey. In particular, adaptive Kalman filtering and weighted RLS approaches can be adopted for dealing with impulsive noises. Oblique projection and instrumental variable-based techniques can handle colored noises. Therefore, it is desirable to extend these approaches from subspace tracking to tensor tracking.

Outliers and Missing Data

They are now becoming more and more ubiquitous in modern datasets. Outliers are data points that appear to be inconsistent with or exhibit abnormal behaviour different from others. Missing observations are often encountered during the data acquisition and collection. Both outliers and missing data can cause several issues (e.g., they introduce bias in estimation) for knowledge discovery from data in general and data streams in particular [6]. Accordingly, dealing with them is an essential task in the analysis of corrupted datasets which has been still a hot topic in data mining for decades. In general, handling such corruptions involves removing/ignoring them after detection or replacing them with alternative values.

There exist few tensor tracking algorithms robust to sparse outliers in the literature. Under the CP format, SOFIA [START_REF] Dongjin | Robust factorization of real-world tensor streams with patterns, missing values, and outliers[END_REF] applies the robust Holt-Winters forecasting model using a pre-cleaning mechanism to identify and down-weight outliers. RACP [27] introduces a ℓ 1 -norm penalty to promote the sparsity on outliers and then uses an ADMM solver to estimate them. Under the 8.2. RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS

Randomized Sketching

It is very well-known that randomized methods can reduce the computational cost of their counterparts while still achieving reasonable estimation [START_REF] Mahoney | Randomized algorithms for matrices and data[END_REF]. Accordingly, many attempts have been made to take their advantages in computation for tensor decomposition in the literature, we refer the readers to [START_REF] Asl | Randomized algorithms for computation of Tucker decomposition and higher-order SVD (HOSVD)[END_REF] for a good overview. Among them, there are a few online algorithms utilizing successfully randomized techniques to speed up the tracking process, such as [29,33,[START_REF] Yang | Incremental PARAFAC decomposition for three-dimensional tensors using Apache Spark[END_REF][START_REF] Ma | Randomized online CP decomposition[END_REF]. Particularly, these algorithms involve solving several overdetermined least-squares (LS) problems. Thanks to the CP and Tucker structures, they use random sampling to build the sampled Khatri-Rao and Kronecker products, and then, recast the original LS problems into randomized ones. Solving the new LS problems can save a lot of computational complexity.

Other randomized techniques (e.g., random projections and count sketch) with other tensor formats have not yet been investigated for tensor tracking and they deserve next investigations in the future.

Parallel and Distributed Computing

The second approach is to develop parallel and distributed computing frameworks for streaming tensor decomposition. It stems from the fact that we can leverage several computational resources to facilitate the tracking process. Moreover, computing systems in a parallel and distributed environment can offer more reliability than their counterparts in a central one as they can avoid the single point of failure which is a fundamental mistake from flaws in the implementation or design of a system. Besides, another appealing advantage of this computing is the scaling upand-out process in which we can add and/or replace computational resources to the system. We refer the readers to [START_REF] Foster | Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering[END_REF] for a good reference.

In the tensor literature, there are several parallel and distributed systems for processing large-scale tensors. We can list here some efficient tools for: (a) distributed CP decomposition (e.g., DFacTo [START_REF] Choi | DFacTo: Distributed factorization of tensors[END_REF], SPLATT [START_REF] Smith | SPLATT: Efficient and parallel sparse tensor-matrix multiplication[END_REF]), (b) distributed Tucker decomposition (e.g., DHOSVD [START_REF] Wang | A distributed HOSVD method with its incremental computation for big data in cyber-physical-social systems[END_REF], SGD-Tucker [START_REF] Li | SGD-Tucker: A novel stochastic optimization strategy for parallel sparse Tucker decomposition[END_REF]), and (c) distributed TT decomposition (e.g., ADTT [START_REF] Wang | ADTT: A highly efficient distributed tensor-train decomposition method for IIoT big data[END_REF], ATTAC [START_REF] Al Daas | Parallel Algorithms for Tensor Train Arithmetic[END_REF]), etc. These tools mainly distribute the unfolding matrices or sub-tensors among several clusters and integrate their low-rank tensor approximations to find the overall low-rank approximation of the underlying tensor. However, most of the existing distributed tensor decompositions are not suitable for handling streaming data. Therefore, it is of great interest to develop practical distributed systems for tracking tensors from data streams.

RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS

Neural Networks-based Methods

Another potential approach is to incorporate neural networks (NNs) into tensor factorization to benefit from their significant advances in computational power. On the one hand, the connection between TDs and NNs has been established in some studies, such as [START_REF] Cohen | Convolutional rectifier networks as generalized tensor decompositions[END_REF][START_REF] Liu | NeuralCP: Bayesian multiway data analysis with neural tensor decomposition[END_REF][START_REF] Wang | Tensor neural network models for tensor singular value decompositions[END_REF]. For example, Cohen et al. in [START_REF] Cohen | Convolutional rectifier networks as generalized tensor decompositions[END_REF] showed that the convolutional NNs with ReLU activation and max/average pooling can be represented by tensor decomposition models. Wang et al. in [START_REF] Wang | Tensor neural network models for tensor singular value decompositions[END_REF] introduced two NN models for finding the low-tubal-rank approximation of three-order tensors. Accordingly, NN tools can be used to model and learn high-order interactions for tensors, and hence, for tensor factorization and tracking. On the other hand, NNs can directly map data streams (temporal slices) as input to the approximation result as output by applying some online learning techniques.

In the literature of machine learning, there exist several kinds of learning capable of dealing with data streams, such as incremental learning, lifelong learning, and online continual learning, to name a few. They can be specifically adapted for tensor tracking.

Others

Next, we present some other issues and problems which also deserve future investigations.

Provable Tensor Tracking

Although the existing tensor tracking methods can provide competitive performance w.r.t. estimation accuracy and/or convergence rate in practice, most of them lack performance guarantees.

The gap between practical uses/implementations and theoretical results in tensor tracking may be caused by the fact that most tensor problems are NP-hard [START_REF] Hillar | Most tensor problems are NP-hard[END_REF], e.g., the best rank-1 tensor approximation is NP-hard even when all observations (temporal slices) are fully observed. Despite several difficulties, there are still attempts to bridge the gap in the literature. Under certain conditions (e.g., the underlying low-rank model remains unchanged over time), some studies established successfully theoretical results to analyse the convergence behavior of their methods, such as [27,29,[START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF][START_REF] Rambhatla | Provable online CP /PARAFAC decomposition of a structured tensor via dictionary learning[END_REF][START_REF]Singleshot: A scalable Tucker tensor decomposition[END_REF]. These initial results encourage us to investigate deeper theory aspects in tensor tracking, such as time variation, asymptotic convergence, and non-asymptotic convergence in low-sample-size settings.

Symbolic Tensor Tracking

In some applications, data may no longer be represented by single (certain) values, but need to be formatted or grouped within sets, intervals, histograms, etc. It leads to the so-called symbolic data analysis (SDA) paradigm in data mining and statistics to deal with such data [START_REF] Brito | Symbolic data analysis: Another look at the interaction of data mining and statistics[END_REF]. In SDA, several new variables types and processing tools have been introduced to represent and analyse 8.2. RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS symbolic data, such as interval-valued, histogram-valued, and categorical modal variables, to name a few. The readers are referred to [START_REF] Brito | Symbolic data analysis: Another look at the interaction of data mining and statistics[END_REF] for a good survey on SDA. In the tensor literature, Mauro et al. in [START_REF] Mauro | Tensor-train decomposition in presence of interval-valued data[END_REF] proposed for the first time a symbolic tensor decomposition for factorizing interval-valued tensors under the tensor-train format. Specifically, the authors extended a set of tools aiming to handle interval-valued matrices for high-order tensors and introduced efficient decomposition and reconstruction strategies. As the symbolic tensor decomposition is in its very early stage of development in both batch and online settings, there are a lot of aspects that need to be investigated in the future.

Tensor Tracking under BTD, t-SVD, Tensor Network formats, and other Variants As reviewed in the sections above, most of the state-of-the-art tensor tracking algorithms are proposed for streaming CP and Tucker decompositions. Despite great success in the batch setting, BTD, t-SVD, and tensor networks (e.g., tensor-train, tensor chain, and tensor ring) have not attracted much attention in real-time stream processing until recently. Thus, developing online methods for tracking tensors under these tensor formats and their variants is essential advantage from their advantage in representing large-scale tensors as well as fulfil the gap between the two most common tensor formats and others. trouver les sous-espaces principaux des facteurs tensoriels [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF].

Décomposition BTD: BTD factorise X en plusieurs blocs de rang multilinéaire faible au lieu de termes rank-1

X BTD = r i=1 G i × 1 U (1)
i × 2 U

(2) Lorsque les observations arrivant à chaque instant sont unidimensionnelles (c'est-à-dire vectorielles), l'intérêt principal de la LRA en ligne est d'estimer le sous-espace principal qui couvre de manière compacte ces observations dans le temps. Plus précisément, on parle de problème de suivi de sous-espace (ST) dans le traitement du signal, qui a été développé pendant plus de trois décennies [START_REF] Comon | Tracking a few extreme singular values and vectors in signal processing[END_REF][START_REF] Delmas | Subspace tracking for signal processing[END_REF][START_REF] Vaswani | Robust subspace learning: Robust PCA, robust subspace tracking, and robust subspace recovery[END_REF]. En général, à l'arrivée des nouvelles données y t ∈ R I 1 ×1 au temps t, la matrice de sous-espace U t ∈ R I 1 ×r peut être dérivé de l'analyse du spectre de la matrice de

i × 3 • • • × N U (

A.2 Description de la Thèse

Conclusion et Perspectives

Le chapitre 8 conclut la thèse avec nos principaux résultats et un apercu des travaux futurs. En particulier, nous présentons plusieurs défis de recherche et problèmes ouverts qui devraient être pris en compte pour le développement du suivi de la composante de rang inférieur des flux de données à l'avenir. Il s'agit de l'imperfection et de la corruption des données; classement et suivi;

1. 1

 1 Internet of Things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 SVD of a rank-r matrix X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Multiway extensions of SVD to high-order tensors: CP/PARAFAC, Tucker, BTD, tensor-train, and t-SVD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Streaming data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Effect of outliers on the standard PCA. . . . . . . . . . . . . . . . . . . . . . . . 1.6 Thesis structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The structure of the survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Adaptive step size η t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Convergence of PETRELS-ADMM in terms of the variation ∥s k+1 -s k ∥ 2 : n = 50, r = 2, 90% entries observed and outlier density ω outlier = 0.1. . . . . . . . . . 3.3 Convergence of PETRELS-ADMM in terms of the variation ∥U t+1 -U t ∥ F : n = 50, r = 2, 90% entries observed and outlier intensity fac-outlier = 10. . . . . . . 3.4 Outlier detection accuracy versus the noise level: n = 50, r = 2, 80% entries observed and 20% outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Outlier detection and data reconstruction: n = 50, r = 2, 90% entries observed, outlier intensity fac-outlier = 1, and outlier density ω outlier = 0.1. . . . . . . . . . 3.6 Impact of outlier intensity on algorithm performance: n = 50, r = 2, 90% entries observed, outlier density ω outlier = 0.1 and SNR = 20 dB. . . . . . . . . . . . . . 3.7 Impact of outlier density on algorithm performance: n = 50, r = 2, 90% entries observed, outlier intensity fac-outlier = 10 and SNR = 20 dB. . . . . . . . . . . . 3.8 Impact of the density of missing entries on algorithm performance: n = 50, r = 2, outlier density ω outlier = 0.1, outlier intensity fac-outlier = 10 and SNR = 20 dB. 3.9 Impact of the corruption fraction by missing data and outliers on algorithm performance: n = 50, r = 2 and fac-outlier = 10 and SNR = 20 dB. . . . . . . . . . . vii LIST OF FIGURES 3.10 Impact of the additive noise on algorithm performance: n = 50, r = 2, 90% entries observed and 10% outliers with intensity fac-outlier = 10. . . . . . . . . . . . . . 3.11 PETRELS-ADMM in time-varying scenarios. . . . . . . . . . . . . . . . . . . . .

4. 1 3 . 4 . 6

 1346 Effect of the forgetting factor β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effect of the noise level σ n on performance of OPIT: sparsity level ω sparse = 90%, time-varying factor ε = 10 -4 , and forgetting factor β = 0.9. . . . . . . . . . . . . 4.3 Effect of the time-varying factor ε on performance of OPIT: sparsity level ω sparse = 90%, noise level σ = 10 -4 , and forgetting factor β = 0.9. . . . . . . . . . . . . . . 4.4 Performance comparisons between OPIT and other SST algorithms in the classical setting: dimension n = 50, snapshots T = 1000, and time-varying factor ε = 10 -3 . 4.5 Performance comparisons between OPIT and other SST algorithms in high dimensions: target rank r = 10, snapshots T = 1000, and time-varying factor ε = 10 -OPITd versus OPIT: Run time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Effect of the target rank r on performance of OPITd: dimension n = 100, snapshots T = 3000, time-varying factor ε = 10 -3 , sparsity level ω sparse = 90%, forgetting factor β = 0.97, and two abrupt changes at t = 1000 and t = 2000. . . 4.8 Effect of the sparsity level ω sparse on performance of OPITd: dimension n = 100, rank r = 20, snapshots T = 3000, time-varying factor ε = 10 -3 , forgetting factor β = 0.97, and two abrupt changes at t = 1000 and t = 2000. . . . . . . . . . . . . 4.9 Four video sequences used in this chapter. . . . . . . . . . . . . . . . . . . . . . . 4.10 Tracking ability of algorithms on the video datasets. . . . . . . . . . . . . . . . . 4.11 OPIT vs the best optimal power-based subspace tracker FAPI: Data dimension n = 100, true rank 10, number of snapshots T = 2000, forgetting factor β = 0.97, abrupt changes at t = 500 and t = 1500. . . . . . . . . . . . . . . . . . . . . . . . 4.12 Performance comparisons between OPIT and other ST algorithms in the classical setting: dimension n = 50, snapshots T = 1000, time-varying factor ε = 10 -3 , and the noise level σ n = 10 -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Internet of Things

1 Y x 2 , 2 Y 3 )

 1223 . . . x 2,M Y 1 Y x N,2 Y . . . x N,M Y Khatri-Rao product (aka Column-wise Kronecker product): Given two matrices Y ∈ R N ×r

1 U

 1 written as X * Y,returns an I 1 × J × I 3 × • • • × I N tensor Z of the recursive form Z = X * Y = fold bcirc(X ) * unfold(Y) ,(1.12) 1.1. BIG DATA STREAM PROCESSING where circ(.) is a block circulant tensor defined as bcirc(I N U I N -1 . . . U 2

Figure 1 . 2 :

 12 Figure 1.2: SVD of a rank-r matrix X.

Figure 1 . 3 :

 13 Figure 1.3: Multiway extensions of SVD to high-order tensors: CP/PARAFAC, Tucker, BTD, tensor-train, and t-SVD.

Figure 1 . 4 :

 14 Figure 1.4: Streaming data.

Figure 1 . 5 :

 15 Figure 1.5: Effect of outliers on the standard PCA.

Figure 1 . 6 :

 16 Figure 1.6: Thesis structure.

[ 25 ]

 25 L. T. Thanh, N. V. Dung, N. L. Trung and K. Abed-Meraim, "Robust Subspace Tracking With Missing Data and Outliers: Novel Algorithm With Convergence Guarantee", IEEE Trans. Signal Process., vol. 69, pp. 2070-2085, 2021. [26] L. T. Thanh, N. V. Dung, N. L. Trung and K. Abed-Meraim, "Robust Subspace Tracking Algorithms in Signal Processing: A Brief Survey", REV J. Elect. Commun., vol. 11, no. 1-2, pp. 15-25, 2021. [27] L. T. Thanh, K. Abed-Meraim, N. L. Trung and A. Hafiane, "Robust Tensor Tracking with Missing Data and Outliers: Novel Adaptive CP Decomposition and Convergence Analysis", IEEE Trans. Signal Process., vol. 70, pp. 4305-4320, 2022. [28] L. T. Thanh, K. Abed-Meraim, N. L. Trung and A. Hafiane, "OPIT: A Simple and Effective Method for Sparse Subspace Tracking in High-dimension and Low-sample-size Context", IEEE Trans. Signal Process., 2022 (submitted).

[ 29 ]

 29 L. T. Thanh, K. Abed-Meraim, N. L. Trung and A. Hafiane, "Tracking Online Low-Rank Approximations of Higher-Order Incomplete Streaming Tensors", Elsevier Patterns, 2022 (submitted). [30] L. T. Thanh, K. Abed-Meraim, N. L. Trung and A. Hafiane, "Streaming Tensor-Train Decomposition With Missing Data", Elsevier Signal Process., 2022 (submitted).

Figure 2 . 1 :

 21 Figure 2.1: The structure of the survey.

2. 4 .

 4 ROBUST SUBSPACE TRACKING IN THE PRESENCE OF IMPULSIVE NOISEreadily incorporated and it can be used as an alternative to constructing the cost function from the sum of square errors like RLS methods. The key idea of APSM stems from that unknown parameters of regression models can be estimated from seeking a point in the intersection of all the sets defined by measurements. In the context of ST, based on the latest observed signals, a cost function is properly chosen at each time instant which scores a zero loss. The next task is to reach the intersection point. To deal with sparse outliers, APSM-type algorithms detect the time instances at which the observed signals are corrupted by outliers via using sparsity-aware

  SNR = 10 dB.

Figure 3 . 2 :

 32 Figure 3.2: Convergence of PETRELS-ADMM in terms of the variation ∥s k+1 -s k ∥ 2 : n = 50, r = 2, 90% entries observed and outlier density ω outlier = 0.1.

Figure 3 . 3 :

 33 Figure 3.3: Convergence of PETRELS-ADMM in terms of the variation ∥U t+1 -U t ∥ F : n = 50, r = 2, 90% entries observed and outlier intensity fac-outlier = 10.

Fig. 3 .

 3 Fig. 3.2 shows the convergence behavior of PETRELS-ADMM w.r.t the two variables: fac-outlier

Fig. 3 .

 3 Fig. 3.4 illustrates the outlier detection performance of PETRELS-ADMM versus the noise level SNR. As can be seen that when we increase the value of SNR from -20 dB to 20 dB, the

1 ,

 1 SNR= 20 dB while varying fac-outlier in the range [0.1, 10]. We can see from Fig. 3.6 that PETRELS-ADMM always outperforms other state-of-the-art algorithms in all testing cases with different fac-outlier 3fac-outlier = 10.

Figure 3 . 4 :

 34 Figure 3.4: Outlier detection accuracy versus the noise level: n = 50, r = 2, 80% entries observed and 20% outliers.

  3.6(a)-(b)). At a high intensity level (e.g. fac-outlier = 5 or 10), PETRELS-ADMM again provides the best performance in terms of both convergence rate and accuracy. GRASTA performs similarly to ReProCS and slightly worse than PETRELS-CFAR (i.e., their SEP values are around 10 -4 ). While ROSETA and NORST fail to recover the underlying subspace in the presence of strong outliers. Note that, in all four experiments above, PETRELS-ADMM always obtains the best SEP value of around 10 -5 and hence is robust to outlier intensity. Impact of outlier density on algorithm performance We fix n = 50, r = 2, 90% entries observed, outlier intensity fac-outlier = 5, SNR = 20 dB while varying the outlier density ω outlier in the range [0.05, 0.4]. The results are shown as in Fig. 3.7.

  3.7(d)). The PETRELS-CFAR works well but has a lower convergence rate and accuracy in terms of SEP metric as compared to PETRELS-ADMM. When the measurement data is corrupted by a smaller number of outliers, PETRELS-ADMM still provides better performance than the others, as shown in Fig.3.7 (a)-(c).

  Outlier detection: SNR = 20 dB.

  Data recovery: SNR = 10 dB.

Figure 3 . 5 :

 35 Figure 3.5: Outlier detection and data reconstruction: n = 50, r = 2, 90% entries observed, outlier intensity fac-outlier = 1, and outlier density ω outlier = 0.1.

Figure 3 . 6 :

 36 Figure 3.6: Impact of outlier intensity on algorithm performance: n = 50, r = 2, 90% entries observed, outlier density ω outlier = 0.1 and SNR = 20 dB.

  ω outlier = 0.4.

Figure 3 . 7 :

 37 Figure 3.7: Impact of outlier density on algorithm performance: n = 50, r = 2, 90% entries observed, outlier intensity fac-outlier = 10 and SNR = 20 dB.

  ωmissing = 0.4.

Figure 3 . 8 :

 38 Figure 3.8: Impact of the density of missing entries on algorithm performance: n = 50, r = 2, outlier density ω outlier = 0.1, outlier intensity fac-outlier = 10 and SNR = 20 dB.

Figure 3 . 9 :

 39 Figure 3.9: Impact of the corruption fraction by missing data and outliers on algorithm performance: n = 50, r = 2 and fac-outlier = 10 and SNR = 20 dB.

  SNR = 5 dB.

Figure 3 . 10 :

 310 Figure 3.10: Impact of the additive noise on algorithm performance: n = 50, r = 2, 90% entries observed and 10% outliers with intensity fac-outlier = 10.

Fig. 3 .

 3 Fig.3.12 shows that the proposed algorithm of PETRELS-ADMM based RMC outperforms GRASTA, LRGeomGC and RPCA-GD. At low outlier intensity (i.e., fac-outlier = 0.1), PETRELS-ADMM based RMC, LRGeomGC and RCPA-GD provide excellent performance even

Figure 3 .

 3 Figure 3.11: PETRELS-ADMM in time-varying scenarios.

Figure 3 . 12 :

 312 Figure 3.12: Effect of outlier intensity on robust matrix completion performance. White color denotes perfect recovery, black color denotes failure and gray colour is in between.

Figure 3 . 13 :

 313 Figure 3.13: Qualitative illustration of video background-foreground separation application.

4 . 4

 44 and Fig.4.5 for illustration.

  taken into account instead of the QR step, OPIT results in a sparse but non-orthogonal mixing matrix U t . The operation requires only O(nr) while the QR step costs a complexity of O(nr 2 ). Therefore, it helps speed up the computation of OPIT especially when r is reasonably high compared to the dimension n. More importantly, with this simple normalization, OPIT can achieve excellent subspace estimation accuracy against the state-of-the-art SST algorithms, please see Figs. 4.4 and 4.5 for examples.

  Nonstationary: ϵ = 10 -3

Figure 4 . 1 :

 41 Figure 4.1: Effect of the forgetting factor β.

Fig. 4 .

 4 Fig. 4.2 and Fig. 7.12 illustrate the effect of the noise level σ n and the time-varying factor ε

Figure 4 . 2 :

 42 Figure 4.2: Effect of the noise level σ n on performance of OPIT: sparsity level ω sparse = 90%, time-varying factor ε = 10 -4 , and forgetting factor β = 0.9.

  n = 1000, r = 50

Figure 4 . 3 :

 43 Figure 4.3: Effect of the time-varying factor ε on performance of OPIT: sparsity level ω sparse = 90%, noise level σ = 10 -4 , and forgetting factor β = 0.9.

  r = 10 , ωsparse = 90%

Figure 4 . 4 :Figure 4 . 5 :

 4445 Figure 4.4: Performance comparisons between OPIT and other SST algorithms in the classical setting: dimension n = 50, snapshots T = 1000, and time-varying factor ε = 10 -3 .

Figure 4 . 6 :

 46 Figure 4.6: OPITd versus OPIT: Run time.

Figure 4 . 7 :

 47 Figure 4.7: Effect of the target rank r on performance of OPITd: dimension n = 100, snapshots T = 3000, time-varying factor ε = 10 -3 , sparsity level ω sparse = 90%, forgetting factor β = 0.97, and two abrupt changes at t = 1000 and t = 2000.

  4.10(a).

  ωsparse = 50%

Figure 4 . 8 :

 48 Figure 4.8: Effect of the sparsity level ω sparse on performance of OPITd: dimension n = 100, rank r = 20, snapshots T = 3000, time-varying factor ε = 10 -3 , forgetting factor β = 0.97, and two abrupt changes at t = 1000 and t = 2000.

Figure 4 . 9 :

 49 Figure 4.9: Four video sequences used in this chapter.

Figure 4 . 10 :

 410 Figure 4.10: Tracking ability of algorithms on the video datasets.

Figure 4 . 11 :

 411 Figure 4.11: OPIT vs the best optimal power-based subspace tracker FAPI: Data dimension n = 100, true rank 10, number of snapshots T = 2000, forgetting factor β = 0.97, abrupt changes at t = 500 and t = 1500.

Figure 4 . 12 :

 412 Figure 4.12: Performance comparisons between OPIT and other ST algorithms in the classical setting: dimension n = 50, snapshots T = 1000, time-varying factor ε = 10 -3 , and the noise level σ n = 10 -1 .

Figure 4 . 13 :

 413 Figure 4.13: Performance comparisons between OPIT and other SST algorithms in high dimensions: target rank r = 10, snapshots T = 1000, time-varying factor ε = 10 -3 , and the noise level σ n = 10 -1 .

Figure 4 .

 4 Figure 4.14: n = 50, T = 200: rank r = 10, time-varying ϵ = 10 -3 , sparsity 90%.

  σn = 10 -3

Figure 4 .

 4 Figure 4.15: n = 1000, T = 500: rank r = 10, time-varying ϵ = 10 -3 , sparsity 90%

Figure 4 .

 4 Figure 4.16: n = 2000, T = 2000: rank r = 20, time-varying ϵ = -3 , sparsity 90%

Figure 4 .

 4 Figure 4.17: n = 5000, T = 2000: rank r = 20, time-varying ϵ = -3 , sparsity 90%

Figure 5 . 1 :

 51 Figure 5.1: Structure of this chapter.

Figure 5 . 2 :

 52 Figure 5.2: Single-aspect and multi-aspect streaming models.

3 ) t U 2 I 1 I 3 I 3 I 3 dFigure 5 . 3 :

 32133353 Figure 5.3: Single-aspect streaming CP decomposition of a third-order tensor.

t- 1 .

 1 Most of the existing BCD-based tracking algorithms suppose that observations are outlier-free (i.e., without O), and hence, they apply the regularized/randomized least-squares 5.4. STREAMING CP DECOMPOSITION methods for solving

Figure 5 . 4 :

 54 Figure 5.4: Multi-aspect streaming CP decomposition of a third-order tensor.

Figure 5 . 5 :

 55 Figure 5.5: Online tensor dictionary learning.

OOOSNBTDO

  aspect streaming method K: iterations, M : number of -Supports side information + columns of side matrices nonnegativity rd 2(m-1) I 2(N -m) -Multi-aspect streaming method d, m: number of coming temporal slices & modes -Adopts SGD + MGS + block tensor dN r N I N -1 + N r 2N -Uses tensor sketching d: (N k + d)I N -Uses tensor random projection d = (s(1 -(s/I) N )/(1 -s/I) -Supports one/two-pass k, s: parameters of projection approximations I N -1 (N Ir + M R + 4M 2 ) -Nonlinear decomposition with M : number of pseudo inputs aFourier features R: size of the pseudo input -Uses Bayesian inference + ADF D-L1-Tucker[START_REF] Chachlakis | Dynamic L1-norm Tucker tensor decomposition[END_REF]

  } and then performs the following steps on each vector y

  : (i) projects it onto the subspace U (n) t-1 , (ii) evaluates the corresponding residual error and the energy for each entry of y

  , the larger the residual error is, the more U (n) t is updated. The complexity of STA is moderate while its effectiveness was demonstrated with the problem of anomaly detection and multi-way latent semantic indexing.In[START_REF] Li | Robust visual tracking based on incremental tensor subspace learning[END_REF][START_REF] Hu | Incremental tensor subspace learning and its applications to foreground segmentation and tracking[END_REF], Hu et al. introduced the so-called IRTSA algorithm to track the dominant 5.5. STREAMING TUCKER DECOMPOSITION subspaces {U (n) t } N -1

  ], Chachlakis et al. proposed a streaming Tucker decomposition called D-L1-Tucker for dealing with streaming tensors. D-L1-Tucker shares the same objective function with ORLTM and OLRTR, but adopts a different approach to handle data corruptions. Particularly on the 5.5. STREAMING TUCKER DECOMPOSITION

Figure 5 . 6 :

 56 Figure 5.6: Online tensor subspace learning.

  a) Augmented Projection. In[START_REF] Baskaran | Accelerated low-rank updates to tensor decompositions[END_REF], Baskaran et al. introduced the so-called LRUT algorithm (which stands for Low-Rank Updates to Tucker decomposition) using a randomized projection technique for tracking the low multilinear-rank approximation of streaming tensors over time. When a data stream arrives, LRUT first projects it onto an extended tensor subspace and then forms an augmented core tensor. Specifically, LRUT adds a few more random dimensions to the current tensor subspace defined by old estimations of the tensor factors. The inclusion of some random vectors here plays a role of noise perturbation aimed to prevent the main optimization from getting stuck in local optima. Next, LRUT performs the standard Tucker decomposition (e.g., batch HOSVD or HOOI) on the resulting augmented core tensor to update tensor factors. In this way, we can avoid the computation of SVD on unfolding matrices of the full tensor which is highly expensive in an online setting. However, its computational complexity is still relatively high since LRUT uses several orthogonalization operations on augmented tensor factors and unfolding matrices of the projected tensor slice. b) Riemannian Optimization. In[START_REF] Kasai | Low-rank tensor completion: A Riemannian manifold preconditioning approach[END_REF], Kasai et al. developed a Riemannian manifold preconditioning approach for tensor completion. Specifically, its stochastic version can be adapted for factorizing incomplete streaming tensors in an online fashion. Since the Tucker format provides an effective representation for tensors in the manifold

5. 5 .

 5 STREAMING TUCKER DECOMPOSITION c) Bayesian Inference. In [255], Fang et al. proposed a Bayesian streaming Tucker decomposition method called BASS-Tucker for handling streaming sparse tensors. Similar to Bayesian methods for streaming CP decomposition, BASS-Tucker adopts the streaming variational Bayes

Stage 1 :

 1 Estimate the coefficient vector u (N ) t

Figure 5 . 7 :

 57 Figure 5.7: Multi-aspect streaming Tucker decomposition of a three-order tensor.

Figure 5 . 8 :

 58 Figure 5.8: Single-aspect streaming tensor-train decomposition.

Figure 5 . 9 :

 59 Figure 5.9: Tracking the rank-(L, L, 1) BTD of 3-rd order streaming X t .

For

  adaptive Tucker decomposition, Kasai and Mishra introduced RPTucker [244], dealing 6.1. INTRODUCTION with dynamic tensor completion. Leveraging a specific Riemannian metric, RPTucker effectively performs preconditioned SGD on the Riemannian manifold of the subspace spanned by tensor factors. Very recently, Gilman and Balzano have proposed TOUCAN (tensor rank-one update on

  In×rn contains the first N loading factors, and u (N ) t

Figure 6 . 1 :

 61 Figure 6.1: Incomplete streaming tensors.

4

 4

(

  A2) Tensor slices {Y t } t≥1 follow the data model (6.47) where the true underlying loading factors U (n) t t≥1 are bounded, i.e., U (n) t F ≤ κ < ∞. When (A1) holds, (A2) naturally holds. It also prevents arbitrarily large values in U (n) t and ill-conditioned computation.

6. 2 .

 2 TENSOR TRACKING WITH MISSING DATA Definition 4. (Leverage Scores & Coherence [323, Definition 2.1]). Given a matrix A = [a ⊤ 1 ;

  |S t |r 2 ) flops from solving the randomized LS regression and forming the sketch for H St . The complexity for updating the loading factor U (n) t comes from the computation of the two matrices ∆Y particular, the first one requires O(|Ω t |r) flops while the latter costs O(I N -2 r 2 ) flops. Note that, the matrix S (n) t

2

 2 flops and reduces to O |Ω t |r + (I N -2 + |S t |)r 2 flops in a parallel scheme. Note that when a preconditioning step (e.g. SRHT) is needed to guarantee the incoherence of H Ωt , ACP requires an additional cost of O |Ω t |r log r flops [328].

n=1=

  r 2 , . . . , r N ], Forgetting factor β, Parameters: α > 0, δ > 0, and m > 0. δI rn .Main Program:Procedure: for t = 1, 2, . . . do

  ii) the loading factors {U (n) } N -1 n=1 and (iii) the core tensor G. The two former estimations are similar to that of ACP, so they require a cost ofO |Ω t |r + (I N -2 + |S 1 |)r2 flops in a parallel scheme where |S 1 | denotes the size of the sampling set of (6.27). The latter estimation costs O |Ω t |r + I N -2 r 2N flops for computing ∆X and ∆G. If using the randomize technique in this stage, the complexity is reduced to O |Ω t |r + |S 2 |r 2N

6. 3 .Figure 6 . 2 :

 362 Figure 6.2: Temporal slice Y t with missing data and sparse outliers.

  Optional) Normalization and Re-estimation of utColumn-wise Normalization:

  ) Stability of Errors: |e t (D t ) -e t-1 (D t-1 )| = O(1/t).6.3. TENSOR TRACKING WITH SPARSE OUTLIERS Proof Sketch. Part (a) can be derived from applying the same arguments of Proposition 11 in our companion work [29]. Parts (b) and (c) are trivial due to the proposed BCD scheme. Part (d) can be obtained by exploiting the Lipschitz continuity and multi-block convexity of the surrogate function ft . We indicate Part (e) by using Part (d) and the Lipschitz continuity of f and f .

Figure 6 . 3 :

 63 Figure 6.3: Effect of the forgetting factor β on the performance of ACP versus the rotation angle α.

= 1 Figure 6 . 4 :

 164 Figure 6.4: Performance of ACP in stationary environments: Y t ∈ R 20×20×20×1000 , the true rank r = 5, an abrupt change at t = 500.

3 Figure 6 . 5 :

 365 Figure 6.5: Convergence behavior of ACP in terms of the objective values f t (U t ) and ∥U t+1 -U t ∥ F .

Figure 6 . 6 :Figure 6 . 7 :

 6667 Figure 6.6: Effect of the noise level σ on the performance of ACP.

Figure 6 . 8 :Figure 6 . 9 :

 6869 Figure 6.8: Tracking ability of four adaptive CP algorithms in a time-varying scenario with 50% missing observations: The tensor of size 20 × 20 × 1000, the noise level σ = 10 -3 , the rotation angle α = π/360 and an abrupt change at t = 600.

Figure 6 . 10 :

 610 Figure 6.10: Performance of ATD versus the missing density ρ and the noise level σ: On the 4-order tensor of size 20 × 20 × 20 × 500 and its Tucker rank r TD = [3, 3, 3, 3].

Figure 6 . 11 :

 611 Figure 6.11: Performance of Tucker algorithms in the case where 50% entries are observed and Tucker rank r TD = [3, 3, 3, 3], and the noise level σ = 10 -2 .

  Completion. In this task, four real video surveillance sequences are used, including Highway, Hall, Lobby and Park 14 . Specifically, Highway contains 1700 frames of size 320 × 240 pixels. Hall has 3584 frames of size 174 × 144 pixels. Lobby consists of 1546 frames of size 128 × 160 pixels. Park includes 600 frames of size 288 × 352 pixels.

Figure 6 . 12 :

 612 Figure 6.12: Effect of the time-varying factor ε on the performance of ATD: Tucker rank [3, 3, 3, 3], 90% entries are observed, the noise level is σ = 10 -2 and an abrupt change at t = 300.

Figure 6 . 13 :

 613 Figure 6.13: Comparison of ATD and ATD-O (orthogonality constraint) in a dynamic scenario: the time-varying factor ε = 10 -2 , the noise level σ = 10 -3 , 70% observations are observed and an abrupt change at t = 300.

Figure 6 . 14 :

 614 Figure 6.14: Effect of the forgetting factor β on the video completion accuracy of ACP and ATC on Lobby data.

( a )

 a Lobby video: 50% missing. (b) Lobby video: Performance of ATD. (c) Hall video: 50% missing. (d) Hall video: Performance of ATD.

Figure 6 . 15 :

 615 Figure 6.15: Performance of adaptive tensor completion algorithms on the video sequences.

Figure 6 . 16 :

 616 Figure 6.16: Waveform-preserving character of ACP on the EEG tensor: 20 channels are missing.

Figure 6 . 17 := 5 Figure 6 . 18 :Figure 6 . 19 :

 6175618619 Figure 6.17: Waveform-preserving character of ACP on the EEG tensor: 40 channels are missing.

Figure 6 . 20 :

 620 Figure 6.20: Impact of outlier intensity (A outlier ) on performance of adaptive CP algorithms; ω miss = 10%, ω outlier = 20%, σ = 10 -2 , ε = 10 -2 .

Figure 6 . 21 :

 621 Figure 6.21: Impact of outlier density (ω outlier ) on performance of adaptive CP algorithms: ω miss = 10%, σ = 10 -2 , ε = 10 -2 , A outlier = 10.

Figure 6 . 22 :

 622 Figure 6.22: Non-Gaussian loading factors.

Figure 6 . 23 :

 623 Figure 6.23: Outlier rejection with different trackers.

Task 1 :Figure 6 . 24 :Figure 6 . 25 :

 1624625 Figure 6.24: Convergence rate of RACP and its modification with the re-update of P t as defined in (6.61): ω miss = 10%, ω outlier = 10%, A outlier = 10, σ = 10 -2 , and ε = 10 -2 .

Figure 6 . 26 :

 626 Figure 6.26: Nonnegative adaptive CP decompositions: Outliers-free, full observations and an abrupt change at t = 600.

6 Figure 6 . 27 :

 6627 Figure 6.27: Experimental results on the Intel Berkeley Lab data.

6 ( 8 Figure 6 . 28 :

 68628 Figure 6.28: Completion accuracy of adaptive CP algorithms on real-world data streams.

Figure 6 . 29 :

 629 Figure 6.29: Epileptic EEG Dataset.

  and RACP. In this experiment, we aim to factorize the EEG tensor into three basis components w.r.t. spatial domain, time-frequency domain, and measurement mode. As there is no real ground truth, we use the results (i.e., CP factors) derived from applying the batch CP-ALS algorithm to the EEG tensor with full observations as benchmarks. Experimental results are shown in Tab. 6.4 and Fig.6.30. They indicate that RACP outperforms NL-PETRELS and provides a slightly better estimation than ACP, especially in the presence of highly incomplete observations (e.g., ≥ 40 channels are missing).

Figure 6 . 30 :

 630 Figure 6.30: First component of EEG factors when 40/60 EEG channels are missing.

Figure 6 . 31 :

 631 Figure 6.31: The error e t over time with α = 1.5 and L t = t. Normal data which are inaccurately labelled as abnormal are referred to as "false positive".

Figure 6 . 32 :

 632 Figure 6.32: Three video surveillance sequences.

Figure 6 . 33 :

 633 Figure 6.33: Qualitative illustration of video background modeling results.

Figure 6 . 34 :

 634 Figure 6.34: Qualitative illustration of video foreground detection results.

6. 6 . 1 . 1

 611 Stage I In order to justify the well-definedness condition, we first indicate that solutions {U t , u (N ) t } ∞ t=1 are bounded and hence obtain several important propositions for the next stages 23 . Proposition 12. Solutions {U t , u t } ∞ t=1 generated by ACP are bounded.

  to Sherman-Morrison formula and the initial caseS (n) 0 = δ n I, S(n)k+1 is a positive definite and invertible matrix and V (n) k+1 is always existent (i.e. inverse of the rank 1 update S (n) k+1 ). In addition, for any positive definite and invertible matrix M ∈ R r×r , we have ∥M∥ F ≤ √ r∥M∥ 2 = √ rσ max (M), and M

  that the proposed RACP algorithm begins with N full-rank matrices U = δ n I, m = 1, 2, . . . , I n .

  , the right hand side of (6.160) is finite, thus u(n) 1,m is bounded for all m. It implies that U (n) 1 is bounded.The induction step: We assume that {U(n) i } ki=1 generated by RACP are bounded at time t = k > 1, we will prove that att = k + 1, U (n) k+1 is also bounded. Since {U (n) k } Nn=1 are assumed to be bounded, u k+1 and W(n) k+1,m are then bounded. In parallel,

  = δI with δ > 0, we obtain S (n) k+1,m ≻ 0, i.e., S (n) k+1,m is nonsingular with the smallest eigenvalue σ min S (n) k+1,m ≥ δ > 0. Thus V (n) k+1,m is always existent. For given M ≻ 0, we always have ∥M∥ F ≤ √ r∥M∥ 2 = √ rσ max (M), and M -1 2 = σ -1 min (M) where σ max (M) and σ min (M) are the largest and smallest eigenvalue of M [9]. Accordingly, we derive ∥V

. 169 )

 169 Proof. The result follows intermediately Theorem 4.1 in[350, page 237].

. 181 )

 181 where f∞ = lim t→∞ ft (.). As indicated in Lemma 1, f∞ D = f ∞ D and hence tr a t V ⊤ ∇f ∞ D ≤ tr a t V ⊤ ∇ f∞ D . Since the above inequality must hold for all V and a t , we obtain tr ∇ f∞ D -∇f ∞ D → 0 a.s. or∇ f∞ D = ∇f ∞ D almost surely. (6.182)As t → ∞, ∇ f∞ D = 0

. 200 )

 200 It is easy to see that the RHS of(6.200) approaches to zero as k → ∞ because of D τ → D and ∇ fk+1 (D τ ) → ∇ f∞ ( D). In parallel, we know that tr[A] -tr[B] = tr[A -B] ≤ √ n∥A -B∥ F and hence tr D ′ -D k ⊤ ∇ fk+1 D k ≤ -ϵ 1 < 0. (6.201) 6.6. APPENDIX

Proposition 25 ( [ 350 ,

 25350 Theorem 4.1, page 237]). Consider a continuous function f : V ×U → R.

  batch settings, TT decomposition has not gained the same popularity in online settings as CP and Tucker decompositions. Particularly, most of the existing TT methods are operating in batch-mode and become inefficient for streaming applications. Related Works: There exist few TT methods related to adaptive tensor decomposition in the literature. In [360-362], Lubich et al. introduced some dynamical tensor approximation methods under TT format for factorizing time-varying tensors, thanks to the Dirac-Frenkel-McLachlan variational principle. However, the dynamical tensors of interest are of fixed size, and hence, their methods indeed belong to the class of batch TT algorithms. In [267], Liu et al. proposed an incremental TT method called iTTD for decomposing high-order tensors of which one dimension grows with time. iTTD factorizes new streams as individual tensors into TT-cores and then

) 7 . 2 .

 72 STREAMING TENSOR-TRAIN DECOMPOSITION Algorithm 9: TT-FOA: First-Order Adaptive Tensor-Train Decomposition

7. 3 .

 3 STREAMING TENSOR-TRAIN DECOMPOSITION WITH MISSING DATA 7.3.2.1 Estimation of the temporal TT-core G (N ) t

  t , [r n-1 , I n , r n ] . The rule(7.40) also suggests that we can incrementally update {G(n) t } N -1n=1 in parallel without disrupting other each. In other words, ATT can support parallel and distributed computing.

7. 4 .

 4 STREAMING TENSOR-TRAIN DECOMPOSITION WITH SPARSE OUTLIERS 7.3.2.3 Complexity Analysis For brevity, we assume that I n = I and r n = r for all n = 1, 2, . . . , N -1. At time t, ATT requires a cost of O(W |Ω t |r 2 ) flops for updating G (N ) t where |Ω t | denotes the number of observed data. Most of operations for updating G (n) t are matrix-matrix products except an inverse operation of a r 2 × r 2 matrix. Thus, ATT requires an extra cost of O (N -1)I N -1 r 4 flops. The overall complexity of ATT is O r 2 max (N -1)I N -1 r 2 , W |Ω t | flops. In term of memory storage, ATT needs O (N -1)(2Ir 2 + r 4 ) words of memory for storing G

Figure 7 . 3 :

 73 Figure 7.3: Temporal slice Y t with missing data and outliers.

Figure 7 . 4 :Figure 7 . 5 :

 7475 Figure 7.4: Effect of the forgetting factor β on the performance of TT-FOA.

Figure 7 . 6 :Figure 7 . 7 :Figure 7 . 8 :

 767778 Figure 7.6: Effect of the time-varying factor σ on the performance of TT-FOA in the case of noise-free.

Fig. 7 . 9 .

 79 Fig. 7.9. In particular, PARAFAC-SDT fails to track video frame while OLCP achieves a worse estimation accuracy than our algorithm. The second task is to demonstrate the effect of TT-rank r TT on the low-rank approximation of the fMRI tensor. The abdominal scans are seen as tensor slices in the online setting. Results of tracking the low-rank component of the last scan are shown in Fig. 7.10. The estimated low-rank fMRI scan deviates from its ground truth when the TT-rank decreases, and hence the relative error increases.

7. 5 Figure 7 . 10 :

 5710 Figure 7.10: Effect of TT-rank on the low-rank approximation of fMRI scans: (a) original MRI scan, (b)-(d) low-rank approximation images for r TT of[START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF],[START_REF] Delmas | Subspace tracking for signal processing[END_REF][START_REF] Delmas | Subspace tracking for signal processing[END_REF] and[START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF][START_REF] He | Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video[END_REF] respectively.

Figure 7 . 11 :

 711 Figure 7.11: Effect of the noise level σ n on the tracking ability of ATT.

Figure 7 . 12 :Figure 7 . 13 :

 712713 Figure 7.12: Effect of the time-varying factor ε on the tracking ability of ATT.

Figure 7 . 14 :

 714 Figure 7.14: The 500-th video frame of "Hall" data: 80% pixels are missing.

7. 5 . 3 . 2 Figure 7 . 15 :

 532715 Figure 7.15: Effect of the noise level σ n on the performance of ROBOT.

Figure 7 . 16 :Figure 7 . 17 :

 716717 Figure 7.16: Effect of the varying factor ϵ on the performance of ROBOT.

Fig. 7 .Figure 7 . 18 :

 7718 Figure 7.18: Effect of the outliers on the tracking ability of ROBOT.

Figure 7 . 19 :

 719 Figure 7.19: Background and foreground separation. From bottom to top row: Highway, Hall, and Lobby. From left to right column: Original video frame, PETRELS-ADMM, GRASTA, and ROBOT.

Figure A. 1 :

 1 Figure A.1: SVD d'une matrice X.

1 k 1 1 G 1 ( 1 ,

 11111 être considéré comme une unification et une généralisation des deux décompositions CP et Tucker bien connues. Plus précisément, lorsque {G i } r i=1 sont des tenseurs diagonaux, BTD se résume à la décomposition CP. Il a la forme d'une décomposition de Tucker lorsqu'un seul terme de bloc (c'est-à-dire r = 1) est considéré. De plus, plusieurs fonctionnalités attrayantes du BTD sont héritées de CP et de Tucker, telles que le calcul stable de Tucker, l'identification et l'unicité de CP[START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF]. En parallèle, il convient de rappeler une remarque dans[START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF] selon laquelle "le rang d'un tenseur d'ordre supérieur est en fait une combinaison des deux aspects : il faut préciser le nombre de blocs et leur taille". Cela signifie que BTD fournit une approche unifiée pour généraliser le concept de rang matriciel aux tenseurs.Décomposition Tensor-Train: Avec (A.1) et (A.4), nous pouvons écrire la SVD de X commeX(i 1 , i 2 ) SVD = r k=1 λ k U(i 1 , k)V(k, i 2 ). (A.7)En conséquence, chaque élément d'un tenseur d'ordre supérieur X peut être représenté parX (i 1 , i 2 , . . . , i N ) TT = r 1 ,r 2 ,...,r N -,k 2 ,...,k N -i 1 , k 1 )G 2 (k 1 , i 2 , k 2 ) . . . G N (k N -1 , i N , 1). (A.8) où G n est un r n-1 × I n × r n tenseur avec n = 1, 2, . . . , N -1 et r 0 = r N = 1.Nous nous référons au modèle de représentation (A.8) en tant que train de tenseurs (TT). Comme CP, le format TT offre un modèle d'économie de mémoire pour représenter les tenseurs d'ordre élevé. Comme Tucker, la décomposition TT et le rang TT r = [r 1 , r 2 , . . . , r N -1 ] de tout tenseur X peuvent être calculés numériquement de manière stable et efficace. Décomposition t-SVD: Enfin, une autre extension de SVD aux tenseurs d'ordre élevé est le tenseur SVD (t-SVD) qui est de la forme suivante: U et V sont des tenseurs unitaires, et G est un rectangle f -tenseur diagonal dont les tranches frontales sont des matrices diagonales [17]. Intuitivement, le modèle t-SVD (A.9) partage la A.1. TRAITEMENT DE FLUX DE DONN ÉES VOLUMINEUSES même forme avec le SVD in (A.1). Cependant, en raison du t-produit " * ", le cadre algébrique utilisé dans le t-SVD est assez différent de l'algèbre (multi)-linéaire classique dans d'autres types de TD et SVD. Par exemple, la plupart des ses calculs sont effectués dans le domaine de Fourier Sous le format t-SVD, le tubal-rank qui est égal au nombre de tubes non nuls de G est utilisé pour définir le LRA des tenseurs de la même manière que le SVD. A.1.2 Approximation de Rang Inférieur en Ligne: Du Sous-espace au Suivi Tensoriel Dans le cadre en ligne, des échantillons de données sont collectés en continu avec le temps. En conséquence, le recalcul des méthodes LRA par lots (par exemple, les algorithmes SVD ou TD par lots) à chaque pas de temps devient inefficace en raison de leur grande complexité et de la variation temporelle, c'est-à-dire de la dérive concept/distribution. Cela a conduit à définir une variante de la LRA appelée LRA en ligne (adaptative) dans laquelle nous pouvons vouloir suivre le processus sous-jacent qui génère des données en continu dans le temps.

Figure A. 3 : 1 ⊞Figure A. 4 :

 314 Figure A.3: Données en continu.

A. 2 . 1 Figure A. 5 :

 215 Figure A.5: Structure de la thèse

  ∥.∥ p , ∥.∥ * Euclidean norm, ℓ p norm, and nuclear norm

	Chapter 1
	General Introduction
	⌊x⌉	integer closest to x
	max{x, y}, min{x, y}	maximum and minimum of x and y
	(.) ⊥	orthogonal (perpendicular) complement

Contents 1.1 Big Data Stream Processing . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Vector, Matrix, and Tensor Operations . . . . . . . . . . . . . . . . . . . 3 1.1.2 Batch Low-rank Approximation: From SVD to Tensor Decomposition . 6 1.1.3 Online Low-rank Approximation: From Subspace to Tensor Tracking . . 10 1.2 Thesis Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Table 1 .

 1 1: Main differences between batch processing and stream processing

	Features	Batch Processing	Stream Processing
	Input	Large batches/chunks of data	(Continuous) streams of data
	Data size	Known and finite	Unknown and/or infinite
	Data type Static	Dynamic/time-varying
	Processing	Process data all at once Process in multiple passes	Process data streams in (near) real time Process in one-or two-pass
	Response	Provide after completion	Provide immediately
	Hardware	Require much storage Require much processing resources Require much less processing resources Require much less storage or no storage
	Time	Take longer time, latencies in minutes to hours	Take a few seconds or faster

1.1. BIG DATA STREAM PROCESSING

Table 2 . 1 :

 21 Surveys on PCA/SE and ST

	Main contribution
	Topic & scope
	Paper

  kinds of non-Gaussian noises (including outliers, impulsive noise, and colored noise) and sparse constraints. Our contributions are as follows. First, in the context of missing data and outliers,

we review four main approaches for dealing with them. They are Grassmannian, recursive least-squares (RLS), recursive projected compressive sensing (ReProCS), and adaptive projected subgradient method (APSM). Second, when the measurements are corrupted by impulsive noise, we show that most of state-of-the-art RST algorithms are based on improving the well-known PAST algorithm which belongs to the class of RLS methods. Two other appealing approaches including weighted RLS and adaptive Kalman filtering are also reviewed. Third, we outline two main classes of RST algorithms that are able to deal with colored noise: instrumental variablebased and oblique projections. Finally, a short review on sparse ST algorithms is presented.

The structure of our review is as follows, please see Fig.

2

.1 for an illustration. Section 2.2 states the problem of RST. In Section 2.3, we provide the state-of-the-art algorithms for the RST problem in the presence of missing data and outliers. The next two sections, 2.4 and 2.5, present RST algorithms that are able to handle impulsive noise and colored noise, respectively. Section 2.6 provides a short review on sparse ST. Finally, Section 2.7 concludes the chapter.

Table 2 .

 2 2.3. ROBUST SUBSPACE TRACKING IN THE PRESENCE OF MISSING DATA AND OUTLIERS 2: Robust subspace tracking algorithms in the presence of both missing data and sparse outliers.

	Algorithm	Approach	Missing Sparse Data? Outliers? Information Start? Guarantee Complexity Prior Warm Convergence Computational

Table 2 .

 2 3: Robust subspace tracking algorithms in the presence of impulsive noise.

	Algorithm	Approach	Burst SIRV α-stable Warm Convergence Computational noise noise noise Start? Guarantee Complexity
	RPAST (2006 [81])	PAST + M-estimation	✓	-	✓	random	✓	O(nr + r 2 )
	MCC-PAST (2014 [82])	Maximum correntropy criterion (MCC) + PAST	✓	-	✓	random	✗	O(nr + r 2 )
	BNC-PAST (2014 [83])	Bounded nonlinear covariance (BNC) + PAST	✓	-	✓	random	✗	O(nr + r 2 )
	robust KFVM Adaptive Kalman filter + (2020 [84]) M-estimation	✓	-	-	random	✗	O(nrℓ + ℓr 2 )+ O(ℓ 2 r + ℓ 3 )
	ROBUSTA (2018 [62])	Weighted RLS + Mahalanobis distance	✓	✓	✓	random	✓	O(nr + r 2 )

Zhang et al. introduced another PAST's variant called MCC-PAST via the maximum correntropy criterion (MCC) in

[START_REF] Zhang | A robust correntropy based subspace tracking algorithm in impulsive noise environments[END_REF][START_REF] Zhang | A robust PAST algorithm based on maximum correntropy criterion for impulsive noise environments[END_REF][START_REF] Zhang | A novel tracking method for fast varying subspaces in impulsive noise environments[END_REF]

. MCC-PAST exploits a correntropy as a new statistic, which can quantify both the time structures and statistics of two random processes, to deal with impulsive noise. Accordingly, the maximum correntropy criterion (MCC) is applied as a substi-2.4. ROBUST SUBSPACE TRACKING IN THE PRESENCE OF IMPULSIVE NOISE ℓ: length of the sliding window -: unknown or undetermined tute for the mean square error criterion in the objective function of PAST. Based on the RLS technique, the MCC-PAST algorithm was then developed. To extend the tracking capability of the MCC-PAS, a variable forgetting factor (FF) technique was also employed in the recursion process. In parallel, Shengyang et al. developed another robust variant of PAST, namely BNC-PAST, to track the underlying subspace via a different criterion [83]. The authors defined a new concept namely bounded non-linear covariance (BNC) to handle relative problems (including ST)

Table 2 .

 2 4: Robust subspace tracking algorithms in the presence of colored noise.

	Algorithm	Approach	Warm Start? Convergence Complexity Guarantee Computational
	IV-PAST (2012 [88])	IV + PAST	random	✗	3nℓ + O(nr)
	IVPM (2014 [89])	IV + propagator-based	random	✗	n(ℓ + 2r)
	LOFF-VR-SREIV-PAST IV + PAST + (2020 [90]) adaptive forgetting factor	random	✓	6nr + 5r 2 + 4n +14r + O(nr)
	obPAST (2005 [91])	Oblique projection + PAST	random	✗	3nr 2 + 3nr + O(r 3 )
	obYAST (2012 [92])	Oblique projection + YAST	random	✗	5nr + O(r 2 + n) + O(r 3 )

ℓ: the dimension of instrumental variable (IV) vector.

Table 2 .

 2 5: Sparse subspace tracking algorithms

	Algorithm	Approach	Prior Information Start? Convergence Complexity Warm Guarantee Computational
	OIST (2016 [93])	Oja method + Soft-thresholding	✗	random	✓	O(nr)
	Streaming SPCA Row truncation (2015 [94]) + QR decomposition	✗	batch	✓	O(nr min(r, s log n))
	ℓ1-PAST (2016 [95])	PAST method + ℓ1-norm sample matrix inverse	✓	random	✗	3nr 2 + 3nr + O(r 2 )
	OVBSL (2017 [96])	Bayesian inference + ℓ2/ℓ1-norm promotion	✓	random	✗	O(nr 2 + nr)
	SS/DS-OPAST (2017 [97])	2-step approach + OPAST + ℓ1-norm approximation	✗	random	✗	3nr 2 + 3nr + O(r 3 )/ 3nr + O(nr 2 )
	SS/GSS-FAPI (2020 [98])	2-step approach + FAPI + Givens rotations	✗	random	✓	2nr 2 + 4nr + O(r 2 )/ 4nr + 4ns + O(r 2 )

2.5. ROBUST SUBSPACE TRACKING IN THE PRESENCE OF COLORED NOISE

  Both obPAST and obYAST minimized a new exponential least-squares cost function where the orthogonal projection in the residual error term is replaced with an oblique one. Experiment results indicate that this modification can facilitate the tracking ability of PAST and YAST in the presence of colored noise. Table2.4 reports further information about these RST algorithms, e.g. convergence and complexity.

Chen et al

. proposed a variant of PAST named oblique PAST (obPAST) to track the signal subspace in

[START_REF] Chen | Subspace tracking in colored noise based on oblique projection[END_REF]

. In the same line, based on the well-known YAST algorithm

[START_REF] Badeau | Fast and stable YAST algorithm for principal and minor subspace tracking[END_REF]

, Florian et al. introduced the new obYAST 2.6. SPARSE SUBSPACE TRACKING algorithm in

[START_REF] Yger | Oblique principal subspace tracking on manifold[END_REF]

.

  : A set of observed signals {x i } t i=1 , x i ∈ R n×1 , observation masks {P i } t i=1 ,P i ∈ R n×n , rank r.

	3.3. PROPOSED PETRELS-ADMM ALGORITHM
	Algorithm 1: Proposed PETRELS-ADMM
	Main Program:
	Procedure:
	for i = 1, 2, . . . , t
	// Estimate outliers s i and coefficient w i using Algorithm 2:
	)

Input

  Input: Observed signals {x i } t i=1 , observation mask P t , the previous estimate U t-1 , forgetting factor β, regularization parameter α, the step size η, ξ t the previous matrix H t-1 .

	3.3. PROPOSED PETRELS-ADMM ALGORITHM
	Algorithm 3: Improved PETRELS for updating U t
	Main Program:
	Procedure:
	.25)

  Convexity of the surrogate functions g t (U)). Given assumptions in Section 3.2.2, the surrogate function g t (U) defined in Eq. (3.6) is not only strongly convex, but also Lipschitz function, i.e., there always exists two positive numbers m 1 and m 2 such that

	3.4. PERFORMANCE ANALYSIS
	Proposition 2 (
		converges to stationary point of L(s, u, r, w, e)
	when k → ∞ and it also implies that the sequence {s k } ∞ k=0 is convergent, i.e.,
	lim k→∞	s k+1 -s k 2 2	= 0.	(3.45)

  as shown in Section 3.3.2. It implies that the estimated U t converges to the stationary point of g t (U).

	Furthermore, since g t (U) is strongly convex and Lipschitz function w.r.t the variable U as
	shown in Proposition 2, we have the following inequality

  5, the convergence rate is fast, i.e. the variation s k+1 -s k

	3.5. EXPERIMENTS
	sequence {U t } t≥0 , generated by PETRELS-ADMM follows the theoretical behavior proved in
	Lemma 2, that is, ∥U t -U t+1 ∥ F	a.s.
		2
	can converge in 50 iterations in both low-and high-noise cases. The results are practical ev-
	idences of Lemma 1. Similarly, Fig. 3.3 shows that the convergence of the variations of the

  9 [131]. The Hall dataset consists of 3584 frames of size 174 × 144 pixels, while the Lobby dataset has 1546 frames of size 144 × 176 pixels. The Sidewalk dataset includes 1200 frames of size 240 × 352 pixels. Highway dataset

	has 1700 frames of size 240×320 pixels. We can see from Fig. 3.13, PETRELS-ADMM is capable
	of detecting objects in video and provides competitive performance as compared to GRASTA Video Frame PETRELS-ADMM PETRELS-CFAR GRASTA
	and PETRELS-CFAR.

  The remaining steps of OPITd require a computational complexity of O(n) only. Accordingly, OPITd costs a complexity of O(r max{n, k log k}) for updating the whole matrix U t at each time t. In practice, we often set the value of k to

O(r log n) or ⌊(1 -ω sparse )n⌉ which is much smaller than n, and thus, the overall complexity of OPITd is approximately linear to nr. OPITd also requires a less memory storage than OPIT. Specifically, its space complexity is 2nr + r for saving U t , S t = [s t,1 , s t,2 , . . . , s t,r ] of size n × r and e t = [e t,1 , e t,2 , . . . , e t,r ] ⊤ of size r × 1 at time t.

  The error matrix ∆C t is bounded in the operator norm with a probability at least 1 -δ:

	4.4. CONVERGENCE ANALYSIS
	Lemma 7.
	.24)
	Proof. See Appendix B.
	Next, Lemma 7 indicates an upper bound on ∥∆C t ∥ 2 which plays a crucial role in Lemma 5
	and 6 as well as establishing the two conditions (4.18) and (4.19) for the convergence of OPIT.

Table 4 .

 4 160 × 1546 174 × 144 × 3584 320 × 240 × 1700 288 × 352 × 600

	applications.

low-sample-size regime. OPIT provides a competitive performance in terms of both subspace estimation accuracy and convergence rate in the classical regime, especially when the SNR level is high. In high dimensions, OPIT outperforms other sparse subspace tracking algorithms, its estimation accuracy is much better than that of the second-best, SS-FAPI. Besides, a fast variant of OPIT has been obtained using deflation called OPITd. Its computational complexity and memory storage are linear to the input size and they are lower than that of OPIT. Simulations carried out on real video sequences indicated that the proposed method has potential for real 1: Runtime and averaged relative error of adaptive algorithms on tracking the four video sequences.

  = S t -Ŝt and κ(S t ) = ∥S # t ∥ 2 ∥S t ∥ 2 , we can bound this distance as follows

	4.7. APPENDIX
	where ∆S t
	2 ,	(4.39)

  2 , C 3 are universal positive parameters, thanks to Proposition 10 and [24, PropositionWe first use proof by induction to prove d t ≤ ω 0 = max{d 0 , ϵ}. Particularly, we already have the base case of d 0 ≤ ω 0 . In the induction step, we suppose d t-1 ≤ ω 0 and then prove d t ≤ ω 0 still holds. After that, we indicate that d t ≤ ϵ is achievable when the two conditions (4.18) and(4.19)

	4.7. APPENDIX
	4.7.4 Appendix D: Proof of Lemma 4
	are met.
	2.1]. As a result, we obtain
	∆C t 2 ≤ c δ σ 2 x	r tW	+ 2σ n σ x + σ 2 n	n tW	,	(4.53)

where c δ = max C 1 , C 2 , C 3 log(2/δ). It ends the proof.

  When d t-1 ≤ ϵ, applying the same arguments in Case 1, we also obtain d t ≤ To sum up, if the two conditions (4.18) and (4.19) are satisfied, then d t ≤ max{d t-1 , ϵ} = ω 0 .

	4.7. APPENDIX
	Case 2: rσ 2 n +ρσ 2 x rξ √ 0 1-ω 2 ϵ ≤ ϵ ≤ ω 0 .
	As a result, the statement d t ≤ ϵ holds if and only if
	rσ 2 n + ρσ 2 x 0 rξ 1 -ω 2	tW	ω 0 ≤ ϵ.	(4.62)
	Specifically, (4.62) is equivalent to
			1/2
			,	(4.61)
	which is exactly the condition (4.19) in Theorem 1. Moreover, there are various options of

p ∈ (0, r] satisfying ρσ 2

x < rξ 1 -ω 2 0 -rσ 2 n , e.g., when the value of ρ is very close to zero. In such cases, d t will decrease in each time t, i.e., d t ≤ d t-1 ≤ ω 0 .

Table 5 .

 5 1: The State-of-the-art Surveys on Tensor Decompositions and Applications

	Class Review (Year) Objects & Topics	Key Contribution
		[185] (1997)	CP/PARAFAC decomposition	An overview of CP decomposition with respect to aspects: features, properties, methods, and applications in chemometrics.
		[186] (2008)	CP & Tucker decomposition
	Surveys on tensor factorization	models, methods, and tools

Table 5 .

 5 2: Main features of the state-of-the-art single-aspect streaming CP decomposition algorithms.

	Algorithm	Missing Data?	Sparse Outliers?	High-order Convergence Warm (N ≥ 4)? Guarantee? Start?	Computational Complexity	Other Information
	PARAFAC-RLST/SDT [211]	✗	✗	✗	✗	✓	O r 2 I 2	-Subspace-based -Tracking using RLST/SDT
	3D-OPAST [212]	✗	✗	✗	✗	✓	O rI 2	-Subspace-based -Tracking using OPAST
	TeCPSGD [106]

  ×r . Accordingly, DTD enables us to divide X t into two parts X t-1 and Y t = X t \X t-1 in order to take advantages

	5.4. STREAMING CP DECOMPOSITION
	from
	U (n) t	N n=1 = argmin
				(n) t-1 with a low cost and then
	estimate the remaining part	Û(n)	(n) t . The tensor factors are particularly derived

of old estimates. We can first update Ū(n)

t incrementally from U t ∈ R d×r of U

Table 5 .

 5 3: Main features of multi-aspect streaming CP decomposition algorithms.

	Algorithm	MAST	OR-MSTC InParTen2 DisMASTD

Table 5 .

 5 which represents a set of T data streams {Y t } T t=1 of the same size I 1 × I 2 × 4: Main features of the state-of-the-art streaming Tucker decomposition algorithms.

	5.5. STREAMING TUCKER DECOMPOSITION
	Algorithm	Missing Data? Outliers? Sparse	High-order Convergence (N ≥ 4)? Guarantee?	Computational Complexity	Additional Information
	STA [236, 237]	✗	✗	✓	✗	O (N -1)rI N -1	-Subspace tracking + deflation
	IRTSA [238, 239]	✗	✗	✗	✗	O 3rI 3 (with N = 3)	-ISVD-based tracking
	ITF [240]	✗	✗	✗	✗	O 3rI 3 (with N = 3)	-ISVD-based tracking
	IHOSVD [241]	✗	✗	✓	✗	O N r 2 I N	-Adopts recursive matrix SVD
						O 3(r + k) 6 I 3	-Adds noise perturbation
	ALTO [242]	✗	✗	✗	✓	k: random columns	-Uses tensor sequential mapping
	LRUT [243]	✗	✗	✓	✗	O N (r + k) 2N I N k: random columns	-Adds noise perturbation -Supports parallel computing
	Riemannian-Tucker	✓	✗	✗	✗	unavailable	-Computes SGD on Riemannian

  , Sun et al. proposed a streaming tensor analysis (STA) algorithm for tracking

	U	(n) t	with time, instead of taking the orthonormal step (5.34b) directly. Particularly on the
	arrival of Y t , STA first divides its unfolding matrix Y	(n) t	into column vectors {y

  can be effectively decomposed into two main stages:(i) estimate the temporal TT-core G under the assumption that Y t is outlier-free. Meanwhile ROBOT adopts an effective ADMM solver to account for the sparse outlier O t . In stage 2, an effective RLS solver was introduced to estimate {G

		(N ) t	and outlier O t , and (ii) update non-temporal TT-
	cores {G	(n) t } N -1 n=1 . In stage 1, TT-FOA and ATT apply the regularized least-squares method
	(N ) to estimate G t

  Shi et al. in [296] developed the so-called STenSr algorithm for anomaly detection and pattern

	discovery in spatio-temporal tensor streams from sensor networks. STenSr utilizes an incremen-
	tal HOSVD and a metric based on Euclidean distance to detect abrupt changes when new
	data comes. Kasai et al. in [297] introduced an online time-structured traffic tensor tracking
	framework to detect network-level anomalies from link indirect measurements over time. In par-
	ticular, it is based on a robust adaptive CP decomposition that uses RLS for tensor tracking
	and ADMM for detecting abnormal flows. Cao et al. in [298] designed an interactive system
	called Voila for detecting and monitoring visual anomalies. Voila is a tensor-based anomaly
	detector with an interaction design that can ranks anomalous patterns based on user input.

Lin et al. in

[START_REF] Lin | Anomaly detection in spatiotemporal data via regularized non-negative tensor analysis[END_REF]

proposed a novel method called TBAD to localize anomalous events. TBAD employs a spatial-feature-temporal tensor model and analyses latent patterns through unsupervised learning. Xu et al. in

[START_REF] Xu | Anomaly detection in road networks using slidingwindow tensor factorization[END_REF]

introduced a tensor-based framework, namely SWTF, capable

  Hu et al. proposed an incremental tensor subspace learning algorithm, called IRTSA, and applied it to robust visual tracking in video streams[START_REF] Hu | Incremental tensor subspace learning and its applications to foreground segmentation and tracking[END_REF]. Li et al. presented a robust algorithm that can update the tensor dictionary and detect anomalies in an online manner, namely RTSL[START_REF] Li | Robust tensor subspace learning for anomaly detection[END_REF]. Sobral et al. introduced an online stochastic tensor algorithm for learning low-rank structure and sparse components in the tensor data[START_REF] Sobral | Online stochastic tensor decomposition for background subtraction in multispectral video sequences[END_REF]. Another incremental tensor decomposition was designed for video background and foreground separation in[START_REF] Kajo | Incremental tensor-based completion method for detection of stationary foreground objects[END_REF]. Li et al.

	developed an adaptive algorithm for robust low-rank tensor learning, called ORLTM [263]. Very

recently, Dimitris et al. have proposed the first robust online Tucker decomposition that can deal with streaming tensors in the presence of outliers

[START_REF] Chachlakis | Dynamic L1-norm Tucker tensor decomposition[END_REF]

. However, none of the above algorithms 6.1. INTRODUCTION are designed for handling missing data. The problem of robust tensor tracking for high-order incomplete streaming tensors remains largely unexplored.

  = diag(vec(P t )), y t = vec(Y t ), and H t has the Khatri-Rao structure, i.e.,

	6.2. TENSOR TRACKING WITH MISSING DATA
	where H t = I N -1 n=1 × n U (n) t-1 . Problem (6.10) can be readily converted into the standard form
	of
	u	(N ) t	= argmin u∈R r	P t y t -H t u	2 2	,	(6.11)
	where P t
							.10)

  t) a.s. (S2) The sequence of nonnegative surrogate values {g t (U t )} ∞ t=1 is quasi-martingale and convergent almost surely. (S3) The empirical loss function {f t (U t )} ∞ t=1 and its surrogate {g t (U t )} ∞ t=1 converge to the same limit, i.e., g t (U t ) → f t (U t ) a.s. Accordingly, {U t } ∞ t=1 converges to a stationary point of f t (U ), i.e., ∇f t (U t ) Details of the analysis is provided in the Appendix A.

t→∞ -→ 0.

  .[START_REF] Shen | Online optimization for max-norm regularization[END_REF] that locally approximates f t (•). Note that ft (D) is not only first-order surrogate, but also a majorant function of f t (D), that is, for all t and D, we always have f t (D) ≤ ft (D) and the error function e t (D) = ft (D) -f t (D) is Lipschitz continuous. In fact, ft (D) and f t (D) converge almost-surely to the same limit, and the solution D

t , which minimizes ft (D), is exactly the one of f t (D) when t → ∞. The results will be later proven in our convergence analysis. In what follows, we propose two solvers for minimizing (6.53) and (6.54) efficiently. 6.3. TENSOR TRACKING WITH SPARSE OUTLIERS Algorithm 8: Robust Adaptive CP Decomposition (RACP)

  = vec(O t ), y t = vec(Y t ), the observation mask matrix P t = diag(vec(P t )), and H t-1 is of a Khatri-Rao structure, i.e., H t-1

	where o t
	t	N )	= argmin o,u	∥o∥ 1 +	ρ 2	P 2 2	,	(6.55)

t y t -o -H t-1 u

Table 6 .

 6 1: Performance of Tucker algorithms on a static 4-order tensor of size 20 × 20 × 20 × 500 and the noise level σ = 10 -2 .

	Rank	[3, 3, 3, 3]	[10, 10, 10, 10]

Table 6 . 2

 62

	2 n ), O t is

: Performance of adaptive tensor decompositions on video data.

Table 6 .

 6 3: Real datasets under the study.

		Dataset	Data size	Tasks
	Intel Berkeley Lab	54 × 4 × 1152	Tracking the online
	Internet Traffic	12 × 12 × 48384	low-rank approximation
	Taxi Trip Record	265 × 265 × 3672	& online data completion
		Hall	176 × 144 × 3584
	Video	Lobby	128 × 160 × 1546	Background modeling & foreground detection
		Highway	240 × 320 × 1700
	EEG	ERPWAVELAB 28 × 64 × 4392 Multichannel EEG analysis & anomaly EEG detection Epileptic data 19 × 500 × 6929

Table 6 .

 6 4: Averaged errors of adaptive CP algorithms for multichannel EEG analysis from incomplete observations. Particularly, we construct an EEG tensor of size 28 × 64 × 4392 (i.e., measurement × channel × time-frequency). To generate incomplete observations, signals from some channels

	Missing channels NL-PETRELS ACP RACP (Proposed)
	1/64	0.051	0.063	0.056
	10/64	0.062	0.025	0.023
	20/64	0.077	0.011	0.014
	30/64	0.121	0.097	0.086
	40/64	0.891	0.132	0.119
	50/64	1.325	1.137	0.982
	signals.

  to obtain the time-frequency representation of multichannel EEG segments (including normal data and seizures), and hence the corresponding EEG tensors of size 19 × 20 × 500 (i.e., channel × scale × time).21 The resulting tensors are then concatenated into a huge tensor whose the last mode is being streamed. We use the first 100 tensors of normal data to obtain a warm start and the estimated rank of 9. Experimental results are shown in Fig.6.31 (the error e t over time) and

	0.8	True Anomaly	False Positive
	0.6
	0.4
	0	200	400	600	800	1000
	Tab. 6.5 (prediction accuracy versus the value of α). Although the results are not really excel-
	lent, it is highly potential to detect anomalies in EEG signals by monitoring the approximation
	error. Subsequent investigations (e.g., type of wavelet, dominant scales, and mother function)

Table 6 .

 6 5: Anomaly EEG detection results. Sensitivity and specificity measure the percentage of anomaly and normal data detected correctly, respectively. Accuracy indicates the overall.To demonstrate the use of RACP for real applications, we consider the problem of video background modeling and foreground detection. Three real video sequences are used in this task, including Hall, Lobby, and Highway 22 (see Fig.6.32). In particular, the Hall video is a set of 3584 images taken at an airport hall, and the image resolution is 176 × 144. The Lobby video contains 1546 images of size 128 × 160 pixels which was captured in an indoor office with switching on/off lights. The Highway video contains 1700 images of vehicles on a highway, and each frame is of size 240 × 320 pixels.

	Value of α Sensitivity Specificity Accuracy
	0.1	42.21%	53.02%	47.57%
	0.5	59.74%	66.48%	63.09%
	1	72.80%	74.38%	73.59%
	1.5	81.58%	85.16%	83.36%
	2	50.16%	53.54%	51.83%
	are necessary to obtain a better prediction.
	Task 3: Video Background-Foreground Modeling

  We then prove that the nonnegative sequence {g t (U t )} ∞ t=1 converges almost surely where {U t } ∞ Proposition 15. Let {U t } ∞ t=1 be a sequence of solutions generated by ACP, the sequence {g t (U t )} ∞ t=1 converges almost surely, i.e.,

	.131)
	6.6.1.2 Step II
	t=1
	is generated by our ACP algorithm.
	Convergence of {g t (U t )} ∞ t=1 can be stated in the following proposition:

∞ t=1 E g t+1 (U t+1 ) -g t (U t )|F t < +∞ a.s.,

  .[START_REF] Lam | Sparsistency and rates of convergence in large covariance matrix estimation[END_REF] Under the given assumptions that variables are bounded, we exploit that the set of loss functions {ℓ(U t , P t , X t )} t≥1 is P-Donsker[START_REF] Van Der | Asymptotic Statistics[END_REF]. As a result, the centered and scaled version off t (U t ) satisfies the following proposition: E √ t f (U t ) -f t (U t ) = O(1), thanks to the Donsker

	6.6. APPENDIX
	theorem [126, Section 19.2].

  .146) where d n is the deterministic positive number. It implies that f t (.) is Lipschitz continuous. Now, we indicate that the nonnegative sequence g t (U t ) -f t (U t )1

	t+1	converges almost
	surely. We prove that the empirical cost function {f t (U t )} ∞ t=1 and its surrogate {g t (U t )} ∞ t=1
	converge to the same limit by showing
	∞
	g t (U t ) -f t (U t ) < +∞.	(6.147)
	t=1

  .155) Because U t is the minimizer of g t (U ), we derive ∇f t (U t ) → 0 a.s. It ends the proof.Boundedness: {D t , O t , u t } ∞t=1 are uniformly bounded.

	6.6. APPENDIX
	6.6.2 Appendix B: Proof of Lemma 11

  Given an incomplete observation P t ⊛ Y t and the past estimation of D, let O t , u * t be the minimizer of l(D, P t , Y t , O, u), i.e.,

	6.6. APPENDIX
	Proposition 17. {u * t , O * t } = argmin u,O	∥O∥ 1 +	ρ 2	P t ⊛ Y t -O -H × N u	2 F .	(6.167)

  .171) It implies that f t (.) is Lipschitz continuous. Since ft (D) and f t (D) are both Lipschitz continuous functions, we then have e t (D t ) -e t-1

  then u t is a quasi-martingale and converges almost surely, i.e., t+1 -u t |F t ] < ∞. , P τ , Y τ , O τ , u τ ) = l(D t , P t+1 , Y t+1 , O t+1 , u t+1 ) , P t+1 , Y t+1 , O t+1 , u t+1 ) -f t (D t ) Since f t (D t ) ≤ ft (D t ) ∀t, we have ft+1 (D t+1 ) -ft (D t ) ≤ l(D t , P t+1 , Y t+1 , O t+1 , u t+1 ) -f t (D t )Define by {F t } t>0 a filtration associated to {u t } t>0 whereF t = {D k , O k , u k } 1≤k≤t recordsall past estimates of RACP at time t. By definition, for every i ≤ t, F i ⊆ F t , and thus, the filtration is interpreted as streams of all historical but not future information generated by RACP. Now, taking the expectation of the inequality (D3) conditioned on F t results inE ft+1 (D t+1 ) -ft (D t )|F t ≤ f (D t ) -f t (D t ) t + 1, where F t is the filtration of past estimations at each time t; the expected cost function f (.) is given by f(D) = lim k→∞ f τ (D), E ℓ(D t , P k+1 X k+1 ) = f (D t ),∀D t and ∀t; and ℓ(D t , P t+1 , Y t+1 ) = l(D t , P t+1 , Y t+1 , O t+1 , u t+1 ) due to {O t+1 , u t+1 } = arg min O,u l(D, P t+1 , Y t+1 , O, u) at time t.

	6.6. APPENDIX
				∞
	t=1 E[u Now, we begin with the following relation when L t = t
	ft+1 (D t ) =	1 t + 1	t+1 τ =1	β t+1-k l(D t t + 1	+	t(β -1) t + 1	ft (D t ) +	t t + 1	ft (D t ).	(6.173)
	Thanks to Lemma 1 and λ ≤ 1, we obtain ft+1 (D t+1 ) ≤ ft+1 (D t ) and
	ft (D t ) -f t (D t ) t + 1	≤ ft (D t ) -ft+1 (D t+1 ) +	l(D t t + 1		.	(6.174)
						t + 1		,	(6.175)

  [START_REF] Chen | Data-intensive applications, challenges, techniques and technologies: A survey on Big Data[END_REF], thanks to the Donsker theorem[START_REF] Van Der | Asymptotic Statistics[END_REF] Section 19.2]. We also deriveAs t → ∞, ft (D t ) → f t (D t ) almost surely We prove {f t (D t )} ∞t=1 and { ft (D t )} ∞ t=1 converge to the same limit by showing , we know that e t (D t )/t + 1 is bounded by ft (D t )-ft+1 (D t+1 ) and (ℓ(D t , P t+1 , Y t+1 )-f t (D t )) (t + 1). Moreover, we have ∞ t=1 ft (D t ) -ft+1 (D t+1 ) < ∞, and the sum of (ℓ(D t , P t+1 , Y t+1 )f t (D t )) (t + 1) also converges due to the convergence of E[f (D t )-f t (D t )] (t + 1) and E[ℓ(D t , P, X )] = f (D t )∀t. Since ∞ ∞ and e t (D t ) -e t-1 (D t-1 ) = O(1/t),we obtain ∞ t=1 ft (D t ) -In what follows, we prove that when t → ∞, ∇ ft D t → ∇f t D t and ∇ ft (D t ) → 0 almost surely. As t → ∞, ∇ ft D t → ∇f t D t almost surely Let D = Ū(1) , Ū(2) , . . . , Ū(N) be the limit point of the sequence of solutions {U

	∞ t=1 we obtain ∞ 1 √ t(t + 1) dt < ∞ after some simple calculations, thus t=1 E δ t E ft+1 (D t+1 ) -ft (D t )|F t < ∞. Therefore, { ft (D t )} ∞ ∞ t 1 √ t(t + 1) < ∞ too. Accordingly, t=1 converges almost surely, i.e., ∞ t=1 E ft+1 (D t+1 ) -ft (D t )|F t < ∞. (6.177) thanks to the quasi-martingale theorem [343, Theorem 9.4 & Proposition 9.5] ∞ t=1 ft (D t ) -f t (D t ) t + 1 < ∞. (6.178) According to (6.174)t=1 1 f t (D t ) < ∞, or ft (D t ) → f t (D t ) a.s., (6.179) thanks to [120, Lemma 3]. 6.6.4 Appendix D: Proof of Lemma 13 t+1 = 6.6. APPENDIX (n) t } t≥1 .

  2

  •••×I N denote the j-th mode-(N + 1) slices of B t . It is easy to see that ft (D) is a multi-block convex and differentiable function and its partial derivative w.r.t. each block is Lipschitz continuous with constant Lt,n = ∥A t,n ∥ F . Accordingly, we have

  in Step 1. Therefore, D is a stationary point of f∞ . In this section, we would provide the following propositions which help us to derive several important results in the proofs. Their details are provided in well-known materials. Proposition 19 ( [132, Section 9.1.2]). The function f is m-strongly convex, with a constant m Proposition 24 ( [120, Lemma 3, page 35]). Let {a t } ∞ t=1 and {b t } ∞ t=1 be two nonnegative sequences such that ∞ i=1

	6.6.5 Appendix E: Useful Propositions

if and only if for all u, v ∈ dom(f ), we always have

|f (v) -f (u)| ≥ m 2 ∥v -u∥ 2 .

Proposition 20

( [132, page 72]

). Every norm on R n is convex and the sum of convex functions is convex.

Proposition 21 ( [132, page 329]). A function f : V → R is called Lipschitz function if there exist a positive number L > 0 such that for all A, B ∈ V, we always have |f (A) -f (B)| ≤ L∥A -B∥. Proposition 22 ( [353, Lemma 1.2.3]). If a function f : V → R is differentiable and its derivative is L-Lipschitz continuous, then for all A, B ∈ V, f (A) -f (B) -∇f (B) ⊤ (A -B) ≤ L

2

∥A -B∥ 2 . Proposition 23. If {f t } t≥1 and {g t } t≥1 are sequences of bounded functions which converge uniformly on a set E, then {f t + g t } t≥1 and {f t g t } t≥1 converge uniformly on E. Proof. Since f t and g t are bounded, we obtain |f t | < M < ∞ and |g t | < N < ∞ for all t. The triangle inequality gives |f t + g t | ≤ |f t | + |g t | < M + N for all t. Also, |f t g t | = |f t ||g t | ≤ M N . Therefore f t + g t and f t g t are bounded.

  6.6. APPENDIXif there exists a constant c such that |ℓ θ 1 (x) -ℓ θ 2 (x)| < c ∥θ 1 -θ 2 ∥ 2 , then F is P-Donsker. For any function ℓ in F , let us define the following functions

Proposition 26 (P-Donsker classes, Donsker theorem

[START_REF] Van Der | Asymptotic Statistics[END_REF] Section 19.2]

). Let F = {ℓ θ : X → R} be a set of measurable functions defined on a bounded subset of R n . For every θ 1 , θ 2 and x,

  .12) where L(.) is a sketching map. Thanks to the Kronecker structure of H[t -1], uniform random sampling can provide a good sketch for H t-1 . Accordingly, we can select rows of H t-1 as well as y t at random to form the sketch H Ω t-1 ∈ R |Ω|×r N -1 and a sampled vector ∈ R |Ω|×1 , where Ω denotes the set of sampling rows. Therefore, g

	(N )
	t

  .I N -1 ×r N -1 are the unfolding matrices of Y t , P t and H t-1 , respectively. Furthermore, (7.29) can be decomposed into W subproblems w.r.t. W

	columns of G (N ) :
		G (N )						2 2	+ λ g i	2 2 .	(7.30)
					-1	H ⊤ t-1 Pt,i y t,i .	(7.31)
	Then, the temporal TT-core G	(N ) t	is simply updated as G	(N ) t	= G	(N ) t-1	G (N ) t	. Note that, we
	can re-update G (N ) t	in the same way above when other TT-cores {G	(n) t } N -1 n=1 are updated.
	7.3.2.2 Estimation of the non-temporal TT-cores G	(n) t	N -1 n=1

t (:, i) = argmin

g i Pt,i y t,i -H t-1 g i

where y t,i = Y t (:, i) and Pt,i = diag{P t (:, i)}. The closed-form solution of the regularized least-squares (7.46) can be given by

G (N ) t (:, i) = H ⊤ t-1 Pt,i H t-1 + λI r N -1

  and the remaining TT-cores, for n = 1, 2, . . . , N -1.

	7.4.2.1 Estimation of the last TT-core G	(N ) t	and Outlier O t
	At each time t, we estimate G (N ) t	and O t by solving

  n) τ while the term O k is discarded due to outlier rejection mechanism (7.47), i.e., P t ⊛ (Y t -O t ) = P t ⊛ Y t . Particularly, (7.48) can be regarded as the optimization problem of adaptive TT decomposition from incomplete observations {Y k } t k=1 with new binary masks { P k } t k=1

Table 7 .

 7 1: Averaged relative error of adaptive tensor decompositions on incomplete video sequences.tensor whose entries are i.i.d. from N (0, σ 2 n ). O t is a sparse tensor containing outliers whose amplitude is uniformly chosen in the interval [0, fac-outlier] while their indices (locations) follow another Bernoulli model with probability ω outlier . L t is the low-rank component of Y t in which g ∈ R r 3 ×1 is a standard normal random vector. At time t, TT-cores are varied under the model G , where ε denotes the time-varying factor, V

			Dataset	Size	Missing	Online Tensor Completion Methods TeCPSGD ACP ATD ATT
			Hall	174 × 144×	×3584	20% 40% 80%	0.1351 0.1412 0.1547	0.1500 0.1366 0.1264 0.1562 0.1370 0.1272 0.1868 0.1472 0.1336
			Lobby	128 × 160×	×1546	20% 40% 80%	0.1307 0.1327 0.1705	0.1320 0.1220 0.1214 0.1375 0.1241 0.1223 0.2142 0.1432 0.1263
			Highway	320 × 240×	×1700	20% 40% 80%	0.2056 0.2119 0.2133	0.2204 0.1980 0.1777 0.2206 0.2001 0.1836 0.2481 0.2089 0.2043
	(4)
	t
		(n) t	= G	(n) t-1 + εV	(n)	(n) t	shares the same
	size as G	(n) t

t

In an adaptive scheme, this subspace might be slowly time-varying, i.e., U = Ut, and hence the adaptive RST algorithm introduced next would not only estimate U but also track its variations along the iterations.

The most direct way of enforcing sparsity constraints is to control the ℓ0-norm of the solution which counts the number of non-zero entries. Following this way, the problem of (3.2) is well specified but computationally intractable. Interestingly, the ℓ1 relaxation can recover the original sparse solution of the ℓ0 problem while still preserving convexity[START_REF] Tropp | Just relax: Convex programming methods for identifying sparse signals in noise[END_REF].

It is referred to as the penalty parameter. Although the convergence rate of the proposed algorithm depends on a specific chosen value, our convergence analysis indicates that the ADMM solver can converge for any positive fixed penalty parameters. However, varying penalty parameters can give superior convergence in practice[114- 117].

Due to the natural ℓ2-ball behavior of the noise (i.e., normal distributed vector) and the sparsity of some unremoved parts of outliers, Huber fitting can be a reasonable choice. The Huber function consists of square and linear terms, so it is less sensitive to variables which have a strong effect on the function ℓ2-norm, but also does not encourage the sparsity like ℓ1-norm.

https://sites.google.com/site/hejunzz/grasta

http://jacarini.dinf.usherbrooke.ca/dataset2012

This phenomenon is often referred to as concept drift or dataset shift in data mining and machine learning[START_REF] Gama | A survey on concept drift adaptation[END_REF].

In an adaptive scheme, the matrix A may be slowly varying with time, i.e., A = At. Our algorithm is capable of successfully estimating the subspace as well as tracking its variation along the time.

With respect to computational complexity, subspace tracking algorithms are categorized into three groups: high complexity O(n 2 r) and O(n 2 ), moderate complexity O(nr 2 ), and low complexity O(nr). The last group, which is referred to as fast algorithms, is the most important class for online processing[START_REF] Delmas | Subspace tracking for signal processing[END_REF].

We limit our analysis in this work to a stationary case when At = A ∀t and β = 1. Establishing the ϵrelative-error approximation guarantee for OPIT in nonstationary environments is non-trivial as data samples do not share the same population. Specifically, finding a tight upper bound on the error matrix ∆Ct -which plays a key role in establishing the two necessary conditions (4.18) and (4.19) as well as Lemmas 1 and 2 -is challenging. Instead of the normal sample covariance matrix (SCM), an exponential weighted variant of the SCM is applied here because of the forgetting factor β < 1. It would make the theoretical convergence analysis more complicated. We leave this challenge for future work.

With respect to the concept of subspace sparsity, Vu et al. in[START_REF] Vu | Minimax sparse principal subspace estimation in high dimensions[END_REF] introduced two notions: column sparsity and row sparsity. Specifically, a subspace is said to be column sparse if some orthonormal basis contains sparse vectors. Meanwhile, every orthonormal basis of a row sparse subspace must consist of sparse vectors. Accordingly, row sparse subspaces also belong to the class of column sparse subspaces. In this work, the proposed OPIT algorithm can achieve an ϵ-relative-error approximation guarantee for the class of column sparse subspaces, and thus, its convergence guarantee also holds under the row sparsity.

For any two orthonormal matrices A and U of the same size, we always have sin θ(A, U) = ∥A ⊤ ⊥ U∥2 = ∥U ⊤ ⊥ A∥2.

In the literature, there exist some other names for the CP decomposition: PARAFAC (Parallel Factors), CPD (Canonical Polyadic Decomposition), and CANDECOMP or CAND (Canonical Decomposition).

Indeed, (A1) is a strong assumption in our analysis, but it can be relaxed as follows: Observed tensor slices {Y t } t≥1 are Frobenius-norm bounded, i.e., ∥Y t ∥F < M < ∞. Low-rank components {Y t } t≥1 of the observed tensor slices {Y t } t≥1 are supposed to be deterministic and bounded. Noise tensors {N t}t≥1 are i.i.d. from a distribution having a compact support.

The value of α can be chosen in the range [10 -3 , 1] for reasonable performance in practice.

In batch setting, the weight vector ut in (6.49) is seen as the t-th row of the last loading factor U (N ) ∈ R I t N ×r of the underlying tensor X t.

The four assumptions (A1)-(A4) are used for the purpose of convergence analysis only, the proposed RACP algorithm can work well in many other scenarios, please see Sec. V for details

Our codes are available at: https://github.com/thanhtbt/tensor_tracking/.

Video sequences: http://jacarini.dinf.usherbrooke.ca/

EEG data: http://www.erpwavelab.org/index.htm

Intel Berkeley Lab: http://db.csail.mit.edu/labdata/labdata.html

Internet Traffic: https://roughan.info/project/traffic_matrix/

Taxi Record: https://www1.nyc.gov/site/tlc/about/tlc-trip-record-data.page

ERPWAVELAB: http://www.erpwavelab.org/

Epileptic EEG Data: https://data.mendeley.com/datasets/5pc2j46cbc/1

As indicated in the EEG dataset description report, data of two channels Cz and Pz were omitted. Thus, we have 19 EEG channels left and each channel contains 500 samples. Also, 20 wavelet scales are chosen in the range[4,[START_REF] Stewart | On the early history of the singular value decomposition[END_REF].

CDNET: http://jacarini.dinf.usherbrooke.ca.

Note that we assume that the underlying tensor slices and their true loading factors are bounded, while in this analysis, we investigate the bound of solutions generated by the proposed ACP algorithm.

https://github.com/thanhtbt/ATT & https://github.com/thanhtbt/ATT-miss & https://github.com/thanhtbt/ROBOT

http://www.changedetection.net/

https://github.com/colehawkins/

Here, the foreground plays the role of outliers and its separation from the background is based on the proposed detection procedure.

Acknowledgments

Appendix E: Additional Experimental Results

OPIT vs the best optimal power-based subspace tracker FAPI Here, we illustrate that OPIT is more effective than the existing power-based subspace trackers.

As it is well-documented that FAPI is the best optimal power-based subspace tracker w.r.t. both convergence rate and estimation accuracy [START_REF] Badeau | Fast approximated power iteration subspace tracking[END_REF], we adopt FAPI in this work. We set the data dimension n = 100, the true rank r = 10, the number of data samples T = 2000. Two levels of noise and time-varying factors are considered, namely σ n = ϵ = 10 -3 and σ n = ϵ = 10 -2 .

To assess how fast subspace trackers converge, we create two abrupt changes at t = 500 and t = 1500. To have a fair comparison, the forgetting factor β is fixed at the same value 0.97 for both OPIT and FAPI in all testing cases. Results are shown as in Fig. 4.11. We can see that OPIT yields higher subspace estimation accuracy than FAPI. When abrupt changes happen, OPIT also converges faster than FAPI.

OPIT vs State-of-the-art Subspace Trackers

In this subsection, we provide further performance comparison of OPIT against the state-of-theart subspace trackers addressed in Section V.4 in the main text. Fig. 4.12 and Fig. 4.13 illustrate the experimental results in the classical regime and high dimensions when the noise level is high, i.e., σ n = 10 -1 . As can be seen that OPIT outperform others completely in both regimes.

Chapter 6

Robust Tensor Tracking with Missing Data and Sparse Outliers Contents

Performance Analysis

In this section, we present a theoretical convergence analysis for the proposed RACP method in Algorithm 1 while assuming D t = D is fixed. Inspired by the recent results of our companion works on robust subspace tracking [25] and tensor tracking [29], we establish a unified theoretical approach to analyse the convergence of the objective values {f t (D t )} ∞ t=1 as well as the solutions {D t } ∞ t=1 generated by RACP.

Assumptions

In order to facilitate the convergence analysis, we make the following assumptions: 11 (A1) Low-rank components {Y t } t≥1 of the observed tensor slices {Y t } t≥1 are supposed to be deterministic and bounded. Entries of noise tensors {N t } t≥1 are zero-mean, independently and identically distributed (i.i.d.) with a small finite covariance, and bounded. Entries of Y t are Frobenius-norm bounded, i.e., ∥Y t ∥ F ≤ M x < ∞, for all t.

(A2) The dictionary D t remains unchanged over time (i.e., D t = D). The loading factors are

Frobenius-norm bounded and the tensor rank r is fixed.

(A3) Observation masks {P t } t≥1 are independent of {Y t } t≥1 , and their entries follow a uniform distribution. The number of observed entries of Y t should be larger than the lower bound O rL log(L) , where L = I 1 I 2 . . . I N . Every row of the mode-n unfolding Y

(n) t of Y t is observed in at least r entries, for n = 1, 2, . . . , N . In addition, each observed entry of Y t is corrupted by outliers independently of others, i.e., the index of outliers is also uniformly random.

(A4) The surrogate function ft (•) is m-strongly multi-block convex, i.e., its second-order derivative w.r.t. each factor is positive-definite, ∇ 2 n ft U (n) , . ⪰ mI ≻ 0 with m > 0.

Among them, assumptions (A1) and (A2) are common for analysing the convergence of online learning algorithms, such as [25,[START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF]. Indeed, (A1) holds in many situations, e.g., real data are often bounded such as audio, image and video. (A2) is a strong assumption as it requires the tensor dictionary to be constant with time. It also prevents arbitrarily large values in U (n) and ill-conditioned computation. Along with (A1), it is interpreted as the simplest possible data model in (robust) tensor tracking where tensor slices are supposed to be generated from a stationary process. Theoretically, stationary processes are often "easier" to model and analyse than nonstationary ones as their statistical properties remain constant over time. Accordingly, stationary has become a common assumption underlying many statistical procedures in general In the following steps, we use the mathematical induction to indicate the bound of U t .

The base case. We prove that the set of solutions

Recall that, the minimizer

1 is derived from the following optimization

for n = 1, 2, . . . , N .

We know that for given M, N ∈ R a×b , ∥M -N∥ F ≥ abs(∥M∥ F -∥N∥ F ) ≥ ∥M∥ F -∥N∥ F .

Accordingly, we have

It is therefore that

where

are initialized by full rank and bounded matrices and

is a full column rank matrix. Under the Assumption (A-3), the null space of

admits only 0 as an element. As a result,

The induction step. Assume that {U i } k i=1 generated by ACP are bounded at time t = k > 1, we will prove that at t = k + 1, U k+1 is also bounded.

where

and a forgetting factor λ ∈ (0, 1] is to discount the effect of past observations. The following steps describe the basic idea of our method for solving (7.4).

Let us denote

, and {G

(n)

t-1 } N n=1 be the old estimated TTcores of X t-1 . Under the assumption that TT-cores are either static or changing slowly, hence H t ≃ H t-1 . Thus, we have

Accordingly, we only need to estimate the last column vector g

instead of re-estimating the whole G (N ) t which becomes inefficient for a large t:

The vector g (N ) t can be updated by minimizing the t-th summand in (7.4):

After that, we update TT-cores

where the two auxiliary tensors are given by

We make the following assumptions for convenience of deploying our method: (A1) TT-cores

n=1 may change slowly between two consecutive instances t -1 and t, i.e.

t-1 ; and (A2) TT-rank vector r TT = [r 1 , r 2 , . . . , r N -1 ] is known and does not change with time.

Proposed Method

In this subsection, we propose an efficient first-order method, namely TT-FOA (which stands for TT adaptive decomposition using First-Order Approach), for tensor-train decomposition of streaming tensors by adapting the alternating minimization framework to the problem (7.44).

The proposed algorithm consists of two main steps: (i) estimate g

Computational Complexity and Memory Storage Analysis

For convenience of the analysis, we assume that the fixed dimensions of the tensor are equal to I while its TT-rank is r TT = [r, r, . . . , r]. In terms of computational complexity, TT-FOA first requires O(|Ω|r 2 ) flops for computing g (N ) t by using the randomized LS method at time t. The cost for updating the k-th TT-core, G (n) t , comes from matrix-matrix products except an inverse operation for S (n) t , hence it costs O(I N -1 r 2 ) flops in general. It is due to that the matrix S (n) t is of size r 2 × r 2 , thus the computation of (S (n) t ) -1 is not expensive and independent of the tensor dimension. Therefore, the overall computational complexity is O(I N -1 r 2 ). In term of memory storage, TT-FOA does not require to save the observation data at each time, it totally costs

When the stochastic TT-FOA is applied, the memory storage is only O (N -1)Ir 2 words of memory.

Streaming Tensor-Train Decomposition with Missing Data

In this subsection, we propose a novel adaptive algorithm called ATT (which stands for Adaptive Tensor-Train) for decomposing high-order incomplete streaming tensors with time under the tensor-train format. By utilizing the recursive least-squares method in adaptive filtering, ATT minimizes effectively a weighted least-squares objective function accounting for both missing values and time-variation constraints on the underlying tensor-train cores. The proposed ATT algorithm is scalable, effective, and technically adept at estimating low-rank components of streaming tensors from noisy, imperfect, and incomplete observations as well as tracking their time variation in nonstationary environments. Besides, ATT can support parallel and distributed computing. To the best of our knowledge, ATT is the first TT algorithm which is capable of dealing with time-dependent streaming tensors with missing values.

Problem Formulation

In this work, we consider the streaming tensor-train decomposition of an N -th order incomplete streaming tensor

We suppose that X t is generated under the following model:

where G (n) ∈ R r n-1 ×In×rn for n = 1, 2, . . . , N with r 0 = r N = 1 is the n-th TT-core (the first and last TT-cores are indeed matrices); [r 1 , r 2 , . . . , r N -1 ] is the TT-rank; and

t } N n=1 can be obtained from:

where Pt is the observation mask of the underlying tensor X t . In online settings, retaking the batch TT methods to solve (7.26) becomes inefficient due to inherent time-variation and nonstationarity of data streams as well as their high complexity in both computation and storage cost. Therefore, we aim to develop a low cost and effective tracker to estimate the TT-cores of

Specifically, we propose to minimize the following exponentially weighted least-squares objective function, instead of (7.26):

where β ∈ (0, 1] is a forgetting factor aimed at reducing the effect of distant observations as well as facilitating the tracking process in dynamic environments; and ρ is a regularization parameter for controlling the time variation of TT-cores between two consecutive instances. Note that, when β = 1 and ρ = 0, the objective function of (7.44) boils down to the batch one of (7.26).

To support our deployment in Section III, we make two mild assumptions on the data model:

n=1 may either be static or vary slowly with time, i.e., G

t-1 ; and TT-rank is supposed to be known.

Proposed Method

In this section, we propose an adaptive method called ATT for adaptive tensor-train decomposition with missing data. Thanks to the block-coordinate descent (BCD) framework, we particularly decompose (7.44) into two main stages: first, update the temporal G (N ) t given old where the 3-way tensor Y t ∈ R I 1 ×I 2 ×I 3 is the t-th slice of X t ; N t is a Gaussian noise tensor of the same size with Y t and ϵ controls the noise level; the last column g

t and G

(3) t are, respectively, of size

where σ controls the variation of the TT-cores between two consecutive instances,

are noise tensors whose entries are i.i.d from the Gaussian distribution with zero-mean and unit-variance.

To measure the estimation accuracy, we use the relative error (RE) metric given by

where X tr (resp. X es ) refers to the true tensor (resp. estimated tensor).

The choice of forgetting factor λ plays a central role in how fast TT-FOA converges. Fig. 7.4

shows the experimental results of applying the algorithm to a static and free-noise tensor whose size is 10 × 12 × 15 × 500 and its TT-rank is r TT = [2, [START_REF] Bahri | Data stream analysis: Foundations, major tasks and tools[END_REF]5]. We can see that the relative error is minimized when λ is round 0.7. TT-FOA fails when λ is close to its infimum or supremum. We then fix λ = 0.7 in the next experiments.

To study the effect of noise on the performance of our algorithm, we vary the value of the noise level ϵ and access its estimation on the same tensor above. The result is shown in Fig. 7.11.

When we reduce the noise, relative error (RE) between the ground truth and estimation degrades gradually and converges towards a steady state error bound. Note that the convergence rate of the algorithm is not affected by the noise level but only its estimation error.

We next consider a scenario where TT-cores change slowly with time and abruptly at instant t = 300. Fig. 7.6 shows the performance of TT-FOA applying to the same free-noise tensor versus the time-varying factor σ. In the same manner to the effect of the noise level, TT-FOA's estimation accuracy goes down when σ increases, but converges towards a steady state error.

Experiment Setup

At time t, the t-th incomplete slice Y t is generated at random under the following model:

t + N t . (7.59)

Here, P t ∈ R I 1 ×I 2 ×I 3 ×1 is a binary tensor whose entries are i.i.d. Bernoulli random variables with probability 1 -ω miss , i.e., ω miss represents the missing density of Y t . Entries of the noise tensor

t ∈ R r 3 ×1 is a Gaussian vector of zero-mean and unit-variance. TT-cores G

t , and G

(3) t are of size I 1 × r 1 , r 1 × I 2 × r 2 , and r 2 × I 3 × r 3 , respectively. Their time variation is modelled as follows G

t , for n = 1, 2, 3, where ε plays a role as the time-varying factor, V (n) t is of the same size as G (n) t and its entries are also i.i.d from N (0, 1).

RESEARCH CHALLENGES, OPEN PROBLEMS, AND FUTURE DIRECTIONS

Tucker format, ORLTM [START_REF] Li | Online robust low-rank tensor modeling for streaming data analysis[END_REF], OLRTR [START_REF] Hu | Streaming data preprocessing via online tensor recovery for large environmental sensor networks[END_REF], and D-L1-Tucker [START_REF] Chachlakis | Dynamic L1-norm Tucker tensor decomposition[END_REF] are able to deal with sparse outliers. Both ORLTM and OLRTR propose to regularize the main objective function with a ℓ 1 -norm regularization. Meanwhile, D-L1-Tucker adopts a threshold-based method to detect outliers. Except for RACP, most of the mentioned algorithms above are not designed for dealing with missing data. In parallel, most of the existing online tensor completion and tracking are sensitive to outliers, such as TeCPSGD [START_REF] Mardani | Subspace learning and imputation for streaming big data matrices and tensors[END_REF], OLSTEC [START_REF] Kasai | Fast online low-rank tensor subspace tracking by CP decomposition using recursive least squares from incomplete observations[END_REF], and ACP [29]. Accordingly, there are plenty of opportunities for us to develop robust tensor tracking from incomplete observations as it is still in its early stage.

Rank Revealing and Tracking

Most of the state-of-the-art tensor tracking algorithms suppose that the tensor rank (e.g., CP, Tucker, TT, or tubal rank) is given as prior information. In practice, it is however a difficult assumption due to the facts that: (i) the tensor rank may change over time and (ii) a good rank determination at the initialization stage is not always guaranteed when the number of training samples is limited and (iii) the exact rank determination may be intractable (e.g., CP rank is NP-hard [START_REF] Hillar | Most tensor problems are NP-hard[END_REF]). Therefore, it is essential to develop tracking algorithms that are capable of revealing the rank over time.

In the literature, there have been many heuristic methods developed for the problem of tensor rank estimation. Most of them adopt the Bayesian approach to infer the tensor rank from data, such as [START_REF] Mørup | Automatic relevance determination for multi-way models[END_REF][START_REF] Bazerque | Rank regularization and Bayesian inference for tensor completion and extrapolation[END_REF][START_REF] Zhao | Bayesian CP factorization of incomplete tensors with automatic rank determination[END_REF]. Theoretically, Bayesian inference offers a good recipe for the tensor rank estimation as we can integrate the low-rank promoting prior as well as the tensor rank into the learning framework. Another possible approach to determine the tensor rank is to use neural networks (NNs), such as [START_REF] Che | Neural networks for computing best rank-one approximations of tensors and its applications[END_REF][START_REF] Zhou | Tensor rank learning in CP decomposition via convolutional neural network[END_REF][START_REF] Hawkins | Towards compact neural networks via end-to-end training: A Bayesian tensor approach with automatic rank determination[END_REF]. Since the rank can be considered as one type of data feature, NNs which can extract hidden features within data can be used to solve the tensor rank determination. Although these methods often require the tensor data to be fully observed, it is possible to readapt or modify them such that their variant are able to handle tensors in an online fashion. For example, we can adopt online Bayesian inference or online learning algorithms for training NNs.

Efficient and Scalable Tensor Tracking

Chapter 5 indicates that most of the existing tensor tracking algorithms are of high complexity.

When we deal with large-scale and high-multidimensional streams, they may become less efficient. Thus, it is necessary to develop efficient and scalable tracking techniques of low cost w.r.t. both computational complexity and memory storage. In what follows, we present three potential approaches which are theoretically capable of accelerating the tracking process, namely (a) randomized sketching, (b) parallel and distributed computing, and (c) neural networks-based methods.

Appendix A

Résumé en Franҫais de la Thèse

par rapport à la norme spectrale et à la norme de Frobenius.

Merci au Théorème 5, la meilleure approximation rang-k de X peut être obtenue en appliquant la procédure suivante:

et la matrice diagonale Λ ∈ R I 1 ×I 2 contient des entrées diagonales positives dans l'ordre décroissant.

• Étape 2 : Sélectionnez les premiers k vecteurs singuliers parmi U et V pour former les matrices

• Étape 3 : Sélectionnez les k valeurs singulières les plus fortes dans Λ pour former:

• Étape 4 : Dérivez la meilleure approximation rang-k de X à partir de:

Lorsqu'il s'agit de tenseurs (aka, tableaux multidimensionnels), plusieurs extensions multivoies du SVD ont été développées pour la décomposition tensorielle (TD) dans la littérature [START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF][START_REF] Cichocki | Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions[END_REF][START_REF] Liu | Tensor Computation for Data Analysis[END_REF]. Les cinq types courants de TD sont CP/PARFAC [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF], Tucker/HOSVD [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF],

tensor-train/network [START_REF] Oseledets | Tensor-train decomposition[END_REF], t-SVD [START_REF] Kilmer | Factorization strategies for third-order tensors[END_REF], et décomposition en termes de blocs (BTD) [START_REF] Lathauwer | Decompositions of a higher-order tensor in block terms-Part II: Definitions and uniqueness[END_REF], voir Décomposition CP/PARAFAC : Semblable à SVD qui représente X par une somme de matrices de rang-1 (c.-à-d. λ i u i v ⊤ i ), la décomposition CP factorise également un tenseur X ∈ R I 1 ×I 2 ו••×I N en termes de rang-1:

(: Décomposition Tucker/HOSVD: En dehors de la forme classique (A.1), on peut exprimer la SVD de X comme suit

En conséquence, une extension multidirectionnelle directe de (A.4) aux tenseurs d'ordre élevé peut être donnée par En conclusion, notre étude apporte plusieurs nouvelles contributions à l'analyse des flux de données massives en général et au problème des LRA en ligne. Il s'agit de nouveaux outils d'analyse permettant de suivre efficacement les flux de données de LRA en ligne, des observations unidimensionnelles aux observations multidimensionnelles dans différents contextes. Par conséquent, ils devraient faire un pas en avant dans les applications en ligne du monde réel.

Mots clés : Flux de données, approximation de rang inférieur, sous-espace, tensoriel.

Signal and Image Data Stream Analytics: From Subspace to Tensor Tracking

Summary: Stream processing has recently attracted much attention from both academia and industry since massive data streams have been increasingly collected over the years. This thesis focuses on investigating the problem of online low-rank approximation (LRA) of data streams over time. When data samples are one-dimensional, the online LRA problem is referred to as subspace tracking. It turns out to be tensor tracking when streaming data are multi-dimensional.

For subspace tracking, we proposed two novel algorithms for tracking the underlying subspace of data streams under two specific scenarios. To deal with sparse outliers and missing data, an effective two-stage subspace tracking algorithm was developed, namely PETRELS-ADMM. The proposed algorithm is based on the alternating direction method of multipliers and recursive least-squares filtering techniques. The second algorithm called OPIT was specifically designed for tracking the sparse principal subspace in high dimensions. Specifically, OPIT introduces a new adaptive variant of power iteration and a new column-based thresholding operator. Both two proposed algorithms belong to the class of provable tracking methods with a convergence guarantee.

For tensor tracking, we developed several new algorithms for tracking the online LRA of streaming tensors over time. Under the CP/PARAFAC format, we leverage the alternative minimization and randomized sketching techniques to develop ACP and RACP which are capable of factorizing incomplete tensors and corrupted tensors, respectively. Under the Tucker format, we proposed another online algorithm called ATD. ATD first tracks the underlying low-dimensional subspaces covering the tensor factors, and then estimates the core tensor using a stochastic approximation. A unified convergence analysis was presented to justify their performance.

In parallel, we designed some adaptive algorithms for streaming tensor-train decomposition which are also capable of tracking the lowrank components of high-order tensors from noisy, imperfect and high-dimensional data with high accuracy.

In conclusion, our study provides several novel contributions to big data stream analytics in general and the online LRA problem in particular. They are new analysis tools allowing to effectively track the online LRA of data streams from one-dimensional to multidimensional observations in different settings, and thus, they are expected to take a step forward real-world online applications.