Florence Achard

Karina Ashurbekova

Sophie Achard

Florence Forbes

Robust structure learning

Keywords: Apprentissage de structure, modèle graphique gaussien, lois de Student, matrice de precision creuse, estimation robuste Structure learning, Gaussian graphical model, t-distribution, sparse precision matrix estimation, robust estimation. 1

published or not. The documents may come

Introduction

Graphs are an intuitive way of representing and visualizing relationships between variables. In a typical graphical model setting, a p-dimensional random vector Y = (Y 1 , Y 2 , • • • Y p ) is represented as an undirected graph denoted by G = (V, E), where V is the set of nodes corresponding to the p variables in Y , and the edge set E describes dependence structure among Y 1 , • • • Y p . In practice, only instances of the variables are observed. The edges encoding the graph structure are unknown and have to be estimated from the observations. Structure or dependence learning has therefore attracted a lot of interest. In a continuous setting, many methods assume the random vector Y is Gaussian with the advantage that for Gaussian vectors, the conditional independence is directly readable from the zeros of the precision matrix. Structure learning can therefore be reduced to standard maximum likelihood parameter estimation. However, in most applications, either the number of observations is too small, or the observations are too noisy, with respect to the dimension of the graph. Most popular approaches face then this issue by resorting to penalized likelihood estimation with the idea of favoring sparse precision matrix estimation. Penalized likelihood approaches based on L 1 -norm of the precision matrix Θ include the work of Banerjee et al. [START_REF] Banerjee | Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data[END_REF] and Yuan and Lin [START_REF] Yuan | Model selection and estimation in the gaussian graphical model[END_REF] while Friedman et al. [START_REF] Friedman | Sparse inverse covariance estimation with the graphical lasso[END_REF] have developed the most popular method -the graphical lasso (glasso), which is a computationally efficient algorithm that maximizes the penalized log-likelihood function through coordinate descent. As an alternative, Cai et al. [START_REF] Cai | A constrained l-1 minimization approach to sparse precision matrix estimation[END_REF] have proposed the CLIME procedure, which is a L 1 minimization technique for precision matrix estimation based on different objective function, that tends to provide more sparse solution than glasso.

The literature on sparse precision matrix estimation is rapidly growing and received significant attention by the research community. Fan et al. [START_REF] Fan | An overview of the estimation of large covariance and precision matrices[END_REF], Cai et al. [START_REF] Cai | Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation[END_REF] gave a nice review of recent results. Many strong methods are valid only for Gaussian variables such as, for example, SCIO suggested by Sun and Zhang [START_REF] Sun | Sparse matrix inversion with scaled lasso[END_REF] and TIGER of Han and Lie [START_REF] Han | Tiger: A tuning-insensitive approach for optimally estimating gaussian graphical models[END_REF]. Although data may deviate from normality in various ways. Outliers and heavy tails frequently occur that can severely degrade the Gaussian models performance. A natural solution is to turn to heavier tailed distributions that remain tractable. In recent papers, Liu et al. [START_REF] Liu | Transelliptical graphical models[END_REF] exploit the connection between Kendall's tau and Pearson's correlation coefficient in the context of transelliptical distributions to obtain robust estimators of correlation matrices. Zhao and Liu [START_REF] Zhao | Calibrated precision matrix estimation for high-dimensional elliptical distributions[END_REF] introduced a strong tool for estimation of sparse precision matrix for elliptical family. This method overcome drawbacks of CLIME algorithms. EPIC simultaneously handles data heavy-tailness and allows to calibrate the regularization for estimating different columns of the precision matrix in CLIME algorithm. It uses the combination of the transformed Kendall's tau estimator and Catoni's M-estimator instead of sample covarince matrix. In the special case of multivariate tdistributions Finegold and Drton [START_REF] Finegold | Robust graphical modeling of gene networks using classical and alternative t-distributions[END_REF] designed a so-called tlasso algorithm to improve graph inference in non Gaussian settings. The tlasso algorithm is based on the use of an Expectation-Maximisation (EM) algorithm that iteratively identifies outlying observations and downweights their impact by using an accordingly modified sample covariance expression. Sparsity is then enforced by replacing the standard M-step by a glasso optimization step.

In this work, we propose to modify the approach of Finegold and Drton [START_REF] Finegold | Robust graphical modeling of gene networks using classical and alternative t-distributions[END_REF] by replacing the glasso M-step with a CLIME M-step, yielding a new procedure for sparse precision matrix estimation referred to as tCLIME. This simple modification enables us to combine benefits from both heavier tail modelling and CLIME optimization. It follows better performance, over tlasso, CLIME and EPIC in estimating the zeros of the precision matrix as illustrated on simulations.

This paper is organized as follows. We first describe briefly glasso and CLIME optimization problems in Section 1. In Section 5, we recall the tlasso algorithm and describe the proposed tCLIME algorithm. Preliminary results on simulations illustrate the superiority of the latter in non Gaussian setting.

Gaussian structure learning

For Gaussian graphical models, conditional independence properties are reflected in the precision matrix Θ = {θ jk } j,k = Σ -1 of the distribution through zero entries (Lauritzen [8]). Thus inferring the graph in this case corresponds to inferring the nonzero elements of the precision matrix. For each nonedge (j, k) ∈ E, Y j and Y k are conditionally independent given all the remaining Y \{j,k} if and only if θ jk = 0.

One of the most commonly used approaches to estimating sparse precision matrix for multivariate normal distribution is through the maximum likelihood. Let Y 1 , • • • , Y n be observed independent samples from N (µ, Σ), Θ = Σ -1 ∈ IR p×p is the precision matrix and S = 1 T is the sample covariance matrix. The negative Gaussian log-likelihood is given by: l(Θ) = tr(SΘ) -log det(Θ). In penalized likelihood methods a L 1 -norm penalty is added to the log-likelihood function to favor zero entries in Θ. One of the most popular techniques to maximize the resulting L 1 -penalized log-likelihood function is glasso (Friedman et al. [START_REF] Friedman | Sparse inverse covariance estimation with the graphical lasso[END_REF]):

n n i=1 (Y i -µ)(Y i -µ)
min Θ 0 L(Θ) = tr(ΘS) -log det(Θ) + ρ Θ 1 ( 1
)
where ρ is a regularization parameter. Larger values of ρ correspond to larger levels of sparsity in Θ. If ρ is equal to zero then the solution of the (1) is the inverse of the sample covariance matrix. An alternative to glasso is a non-likelihood based approach called CLIME which solves instead:

min Θ 1 s.t.: |SΘ -I| ∞ ρ, (2)
where ρ is a tuning parameter and S is the sample covariance matrix. No symmetry condition is imposed on Θ and symmetry is achieved by setting both estimates θ ij and θ ji to the one with the smaller magnitude. Note that theoretical justification of CLIME algorithm heavily relies on the Sub-Gaussian tail assumption. However CLIME, as glasso, still uses the usual sample covariance matrix as input which makes it sensitive to outliers. As explained in Cai et al. [START_REF] Cai | A constrained l-1 minimization approach to sparse precision matrix estimation[END_REF], the CLIME motivation and its relationship to glasso comes from the following observation. The solution of the glasso optimization problem:

Θ glasso = min Θ 0 tr(ΘS) -log det(Θ) + ρ Θ 1 (3)
satisfies ( Θ glasso ) -1 -S = ρ Z, where Z is an element of the subdifferential ∂ Θ glasso 1 . This leads to the optimization problem:

min Θ 1 s.t.: |Θ -1 -S| ∞ ρ, (4)
After multiplying the constraint with Θ we find the problem (2).

T-distribution graphical models: tlasso and tCLIME

Among the various existing forms of the multivariate t-distribution (Nadarajah and Kotz [START_REF] Nadarajah | Multivariate t Distributions and their Applications[END_REF]), the most common form with parameters µ, Σ and ν is given by:

t p (y; µ, Σ, ν) = Γ( ν+p 2 ) (πν) p/2 Γ( ν 2 )|Σ| 1/2 1 + δ(y, µ, Σ) ν -(ν+p)/2
where δ(y, µ, Σ) = (y -µ) T Σ -1 (y -µ) is the Mahalanobis distance between y and µ, y ∈ R p . The vector µ ∈ R p and the positive definite matrix Σ determine the first two moments of the distribution when ν > 2. The covariance matrix of the t-distribution is ν/(ν -2)Σ. We will keep the convenient notation of Finegold and Drton [START_REF] Finegold | Robust graphical modeling of gene networks using classical and alternative t-distributions[END_REF] and denote by Θ = Σ -1 to draw a parallel between Gaussian graphical models and graphical models based on t-distribution. Unlike the Gaussian case, in a t-distribution θ jk = 0, j = k no longer corresponds to conditional independence of Y j and Y k given all the remaining Y \{j,k} . However, despite the lack of conditional independence, the conditional uncorrelation property still holds. Mimicking the glasso idea, Finegold and Drton [START_REF] Finegold | Robust graphical modeling of gene networks using classical and alternative t-distributions[END_REF] proposed to add a L1-norm penalty on Θ to the maximization of the log-likelihood function in the t-distribution case. It follows a tlasso procedure that is more robust to outliers. Inference is made using an EM algorithm: τ is a hidden variable and conditional distribution of Y given τ is N p (µ, Σ/τ ). Degree of freedom parameter ν is assumed to be known. At the (t + 1)th M-step, an update value Θ (t+1) is found by solving the following optimization problem:

min Θ 0 tr(ΘS τ (t+1) Y Y (µ (t+1) )) -log |Θ| + ρ Θ 1 , (5)
where τ (t+1) i

is the conditional expectation of τ i calculated in the (t + 1)st E-step as τ (t+1) i = ν+p ν+δ Y (µ (t) ,Σ (t) ) . A "weighted sample covariance matrix" is then computed as:

S τ (t+1) Y Y (µ) = 1 n n i=1 τ (t+1) i (Y i -µ (t+1) )((Y i -µ (t+1) )) T (6)
and

µ (t+1) = n i=1 (τ (t+1) i Y i )/ n i=1 τ (t+1) i
. Note that ( 5) is a similar objective function minimized by glasso, so that the proposed tlasso reduces to solving at each iteration a glasso optimization.

In the same EM framework, we propose to replace glasso by CLIME for its better performance in the M-step. It follows that (5) becomes:

min Θ 1 s.t.: |S τ (t+1) Y Y (µ (t+1) )Θ -I| ∞ ρ, (7)
We refer to the resulting procedure as tCLIME. As tlasso, tCLIME uses the "weighted sample covariance matrix" instead of the sample covariance matrix, which makes both methods less sensitive to outliers in comparison to their Gaussian counterparts (glasso and CLIME).

Simulation results

Since tCLIME is a combination of tlasso and CLIME, the tCLIME performance is compared to that of tlasso and CLIME. We also compare tCLIME with the EPIC method which is a strong estimator of sparse precision matrix for elliptical distributions. We generate a random 100 × 100 sparse precision matrix Θ according to the procedure described in Finegold and Drton [START_REF] Finegold | Robust graphical modeling of gene networks using classical and alternative t-distributions[END_REF] and simulate n=150 observations from t 100 (0, Θ -1 , 3) and N 100 (0, Θ -1 ). The four algorithms are then run with different values of ρ. To measure how well the sparsity of the true precision matrix is recovered, the whole process is repeated 50 times. The corresponding ROC curves are shown in Figure 1. For Gaussian data, the tCLIME and EPIC performance is similar and better than that of tlasso and CLIME. When data are generated from a t-distribution, tCLIME significantly outperforms CLIME and also shows better results than EPIC and tlasso.

Conclusion

We have introduced tCLIME a modified version of the tlasso algorithm. tCLIME provides competitive results and can be used for robust sparse precision matrix estimation. However further simulations should be made to account for higher dimensional cases, as well as tests on real data.

Bibliographie

Figure 1 :

 1 Figure 1: ROC curves illustrating the performance of tlasso, CLIME, tCLIME and EPIC methods on 2 different data sets. The tuning parameter ρ is chosen in the range [0.1, 2.5] with stepsize 0.05 for tlasso and [0.01, 0.4] with stepsize 0.01 for CLIME, tCLIME anf EPIC.


Dirichlet process (DP) Spatially-constrained mixture model: DP-Potts mixture model Finite mixture model Bayesian finite mixture model DP mixture model DP-Potts mixture model Inference using variational approximation Some image segmentation results Conclusion and future work

Dirichlet process (DP)

The DP is a central Bayesian nonparametric (BNP) prior1 .

Definition (Dirichlet process)

A Dirichlet process on the space Y is a random process G such that there exist α (concentration parameter) and G 0 (base distribution) such that for any finite partition {A 1 , . . . , A p } of Y, the random vector (P (A 1 ), . . . , P (A p )) will be Dirichlet distributed:

(P (A 1 ), . . . , P (A p )) ∼ Dir(αG 0 (A 1 ), . . . , αG 0 (A p ))

Notation: G ∼ DP(α, G 0 )
The DP is the infinite-dimensional generalization of the Dirichlet distribution.

A DP prior G can be constructed using three methods:

The Blackwell-MacQueen urn scheme

The

y i = (y i1 , ..., y iD ) ∈ R D i.i.d p(y i |θ * , π) = K k=1 π k F (y i |θ * k )
where θ * = (θ * 1 , ..., θ * K ) and π = (π 1 , ..., π K ) with θ * class parameters and π mixture weights with K i=1 π i = 1. θ * and π can be estimated using EM algorithm.

Equivalently

G = K k=1 π k δ θ * k non random θ i ∼ G and y i |θ i ∼ F (y i |θ i ).

Bayesian finite mixture model

In a Bayesian setting, a prior distribution is placed over θ * and π.

Thus, the posterior distribution of parameters given the observations is

p(θ * , π|y) ∝ p(y|θ * , π)p(θ * , π)
To generate a data point within a Bayesian finite mixture model:

θ * k ∼ G 0 π 1 , ..., π K ∼ Dir(α/K, ..., α/K) G = K k=1 π k δ θ * k is a random measure θ i |G ∼ G, which means θ i = θ * k with probability π k y i |θ i ∼ F (y i |θ i )
Bayesian finite mixture model In a Bayesian setting, a prior distribution is placed over θ * and π.

Thus, the posterior distribution of parameters given the observations is

p(θ * , π|y) ∝ p(y|θ * , π)p(θ * , π)
To generate a data point within a Bayesian finite mixture model:

θ * k ∼ G 0 π 1 , ..., π K ∼ Dir(α/K, ..., α/K) G = K k=1 π k δ θ * k is a random measure θ i |G ∼ G, which means θ i = θ * k with probability π k y i |θ i ∼ F (y i |θ i ) Limitation:
Require specifying the number of components K beforehand.

Solution:

Assume an infinite number of components using BNP priors. Spatial constraints and dependencies are not considered.

DP mixture model

G ∼ DP(α, G 0 ) θ i |G ∼ G y i |θ i ∼ F (y i |θ i )

Solution:

Combine the DP prior with a hidden Markov random field (HMRF).

DP-Potts mixture model

To solve the issue, we introduce a spatial Potts model component:

M (θ) ∝ exp   β i∼j δ z(θi)=z(θj )  
with θ = (θ 1 , ..., θ N ) and β the interaction parameter.

The DP mixture model is thus extended:

G ∼ DP(α, G 0 ) θ|M , G ∼ M (θ) × i G(θ i ) y i |θ i ∼ F (y i |θ i ) DP-
G ∼ DP (α, G 0 ) θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ), k = 1, 2, . . . G = ∞ k=1 π k (τ )δ θ * k + θ i |G ∼ G y i |θ i ∼ F (y i |θ i ) = Dirichlet
θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ), k = 1, . . . G = ∞ k=1 π k (τ )δ θ * k =⇒ θ i |G ∼ G y i |θ i ∼ F (y i |θ i ) Stick breaking construction of DPMM θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ), k = 1, . . . θ i = θ * k with probability π k (τ ) y i |θ i ∼ F (y i |θ i )
DP-Potts mixture model

DP-Potts: Stick breaking construction

Using assignment variables z i DPMM view

θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ), k = 1, . . . θ i = θ * k with probability π k (τ ) y i |θ i ∼ F (y i |θ i ) =⇒ Mixture/Clustering view θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ), k = 1, . . . p(z i = k|τ ) = π k (τ ) with z i = z(θ i ) = k when θ i = θ * k y i |z i , θ * ∼ F (y i |θ * zi )

DP-Potts: Stick breaking construction

Using assignment variables z i

Stick breaking of DPMM

θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ) p(z i = k|τ ) = π k (τ ) y i |z i , θ * ∼ F (y i |θ * zi )
Stick breaking of DP-Potts

θ * k |G 0 ∼ G 0 τ k |α ∼ B(1, α), k = 1, 2, . . . π k (τ ) = τ k k-1 l=1 (1 -τ l ) p(z|τ, β) ∝ i π zi (τ ) exp(β i∼j δ zi=zj ) z = {z1, . . . , z N } y i |z i , θ * ∼ F (y i |θ * zi )
NB: Well defined for every stick breaking construction ( Access the intractable p(Z, Θ|y, Φ) approximate as q(z, Θ) = q z (z)q θ (Θ)

Variational Expectation-Maximization

Alternate maximization in q z and q θ (φ are hyperparameters) of the Free Energy: Quantitative evaluation of the segmentations A general DP-Potts model and the associated VB algorithm were built.

F(q z , q θ , φ) = E qzq θ log p(y, Z, Θ|φ) q z (z)q θ (Θ) = log p(y|φ) -KL(q z q θ , p(Z, Θ|y, φ)) DP-Potts Variational EM procedure Joint DP-Potts (Gaussian) Mixture distribution p(y, z, τ , α, θ * |φ) = N j=1 p(yj|zj, θ * ) p(z|τ , β) ∞ k=1 p(τ k |α) ∞ k=1 p(θ * k |ρ k ) p(α|s1, s2) p(yj |zj , θ * ) = N (yj |µz j , Σz j ) is Gaussian p(z|τ , β) is a DP-Potts model p(τ k |α) is Beta B(1, α) p(θ * k |ρ k ) = N IW(µ k , Σ k |m k , λ k , Ψ k , ν k ) is Normal-inverse-Wishart p(α|s1, s2) = G(α|s1, s2) is Gamma Usual truncated variational posterior, qτ k = δ1 for k ≥ K (eg. K = 40) q(z, Θ) = N j=1 qz j (zj) qα(α) K-1 k=1 qτ k (τ k ) K k=1 q θ * k (µ k , Σ k ) E-steps: VE-Z, VE-α,
The DP-Potts model was applied to image segmentation and tested on different types of datasets.

Impact of the interaction parameter β on the final results is significant.

An estimation procedure was proposed for β

A general DP-Potts model and the associated VB algorithm were built.

The DP-Potts model was applied to image segmentation and tested on different types of datasets.

Impact of the interaction parameter β on the final results is significant.

An estimation procedure was proposed for β How does β influence the number of components?

Extend the model with other priors (Pitman-Yor process, Gibbs-type priors, etc.).

p(z|τ , β) = K(β, τ ) -1 exp(V (z; τ , β)) with V (z; τ , β) = i log πz i (τ ) + β i∼j δ (z i =z j ) β = arg max β Eq z qτ log p(z|τ ; β) = arg max β Eq z qτ V (z; τ , β) -Eq τ log K(β, τ )
Two difficulties

(1) p(z|τ , β) is intractable (normalizing constant K(β, τ ), typical of MRF)

(2) it depends on τ (typical of DP) Two approximations

(1) "standard" Mean Field like approximation a exp(log π l (τ ) + β i∈N (j) q zi (l))

and τ = E qτ [τ ]
β is estimated at each iteration by setting the approximate gradient to 0

E qzqτ ∇ β V (z; τ , β) = K k=1 i∼j
q zj (k) q zi (k)

∇ β E qτ log K(β, τ ) = E p(z|τ ,β)qτ ∇ β V (z; τ , β) ≈

∼

  Chinese Restaurant Process The Stick-Breaking construction Dirichlet process (DP) construction A DP prior G can be constructed using three methods: The Blackwell-MacQueen urn scheme The Chinese Restaurant Process The Stick-Breaking construction The DP has almost surely discrete realizations 2 : G 0 and π k = πk l<k (1 -πl ) with πk iid ∼ Beta(1, α).

  mixture model: DP-Potts mixture Clustering/segmentation: Finite mixture models assume data are generated by a finite sum of probability distributions: y = (y 1 , ..., y N ) with

Froma

  Bayesian finite mixture model to a DP mixture model To establish a DP mixture model, let G be a DP prior (K → ∞), namely G ∼ DP(α, G 0 ) and complement it with a likelihood associated to each θ i To generate a data point within a DP mixture model:

  . Pitman-Yor (τ k |α, σ) ∼ B(1 -σ, α + kσ) Inference using variational approximation Clustering/ segmentation task: Estimating Z while parameters Θ unknown , eg. Θ = {τ , α, θ * } Bayesian setting

  VE-τ and VE-θ * M-step: φ updating straightforward except for β Model validation and verification: Segmented image using DP-Potts model with β = 2.5. Segmentation with estimated β = 1

( 2 )

 2 Replace the random τ by a fixed τ = E qτ [τ ] a Forbes & Peyrard 2003 Approximation of p(z|τ ; β) p(z|τ , β) ≈ qz (z|β) = N j=1 qzj (z j |β)qzj (z j = k|β) = exp(log π k (τ ) + β i∈N (j) q zi (k)) ∞ l=1

  Process Mixture Model (DPMM)

	Spatially-constrained mixture model: DP-Potts mixture model	DP-Potts mixture model
	DP-Potts: Stick breaking construction
	Stick breaking construction of DPMM
	JSM 2018	July 2018	13 / 35

H.

Lü et al.

  Probabilistic Rand Index on 154 color (RGB) images with ground truth (several) from Berkeley dataset (1000 superpixels). But Manual ground truth segmentations are subjective !

	PRI results with DP-Potts model
		Mean	Median	St.D.
	K=10	71.48	72.54	0.1040
	K=20	73.64	73.42	0.0935
	K=40	75.33	76.47	0.0853
	K=50	75.81	76.31	0.0873
	K=60	76.55	77.12	0.0848
	K=80	77.06	78.30	0.0835
	PRI results from Chatzis 2013

Computation time : Berkeley 321x481 image reduced to 1000 superpixels takes 10-30 s on a PC with CPU Intel(R) Core(TM) i7-5500U CPU 2.40GHz and 8GB RAM

Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. The Annals of Statistics, 1(2):209-230.H. Lü et al.JSM

July 2018

Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4:639-650. H. Lü et al. JSM 2018 July 2018

[START_REF] Lü | [END_REF]

2D point clustering (unsupervised learning) based on the DP mixture model: Let the data speak for themselves! H. Lü et al. JSM 2018 July 2018

DP-Potts mixture model

[START_REF] Orbanz | Nonparametric Bayesian image segmentation[END_REF];[START_REF] Xu | Bayesian nonparametric image segmentation using a generalized Swendsen-Wang algorithm[END_REF];[START_REF] Sodjo | A generalized Swendsen-Wang algorithm for Bayesian nonparametric joint segmentation of multiple images[END_REF]

[START_REF] Albughdadi | A Bayesian non-parametric hidden Markov random model for hemodynamic brain parcellation[END_REF]

Chatzis & Tsechpenakis (2010); Chatzis (2013) H. Lü et al. JSM 2018 July 2018

Segmentation results for SAR images: Original image Segmentation by DP-Potts (K=40, β = 0) Segmentation by DP-Potts (K=40, β = 2) Segmentation by DP-Potts (K=40, β = 10) Original image Segmentation by DP-Potts (K=40, β = 0) Segmentation by DP-Potts (K=40, β = 2) Segmentation by DP-Potts (K=40, β = 10)

The segmentation results obtained by DP-Potts model with β = 0, 1, 5.

Job opportunity

Université Grenoble Alpes invites applications for a 2-year junior research chair (post-doc) Variational EM

General formulation, at iteration (r)