Keywords: sériation, méthodes spectrales, optimisation combinatoire, relaxations convexes, permutations, permutaèdre, optimisation robuste, assemblage de novo, séquençage de troisième génération, Oxford Nanopore Technology, Overlap-Layout-Consensus, classement. ii seriation, spectral methods, combinatorial optimization, convex relaxations, permutations, permutahedron, robust optimization

Les technologies de séquençage d'ADN ne permettent de lire que de courts fragments, dont on ignore la position sur le génome. L'assemblage de novo vise à reconstituer une séquence d'ADN entière en mettant ces fragments bout-à-bout, tel un puzzle. Dans l'approche OLC (overlap-layout-consensus), on calcule le chevauchement entre fragments afin de les disposer en ordre (réarrangement ), puis extraire une séquence consensus.

Le réarrangement peut s'écrire comme un problème combinatoire de sériation, où l'on réordonne des éléments comparable entre eux, de sorte que deux éléments adjacents sont similaires. Ce problème est résolu efficacement par un algorithme spectral en l'absence de bruit, mais il en va autrement des données génomiques réelles. En particulier, des régions du génome sont similaires bien qu'éloignées (séquences répétées), rendant l'assemblage problématique.

Les méthodes d'assemblage emploient des algorithmes hiérarchiques et gloutons pour désambiguïser les séquences répétées. Nous proposons ici une approche épurée où l'on réarrange tous les fragments «d'un coup» via la résolution de sériation.

Notre première contribution montre que l'emploi de la méthode spectrale pour le réarrangement s'intègre parfaitement dans le schéma OLC, produisant des résultats de qualité semblable aux méthodes standard. Cependant, du fait des séquences répétées, cette méthode produit des assemblages fragmentés (typiquement en quelques sous-séquences au lieu d'une).

La deuxième contribution est un prolongement de la méthode spectrale lié à la réduction de dimension sous conservation de distances, englobant les problèmes de sériation et de sériation circulaire (une variante où les éléments peuvent être ordonnés selon un cycle) dans un cadre unifié. Ce prolongement rend l'algorithme robuste au bruit et résout le problème de fragmentation de l'assemblage précédent.

Notre troisième contribution formalise la sériation robuste, où l'on souhaite réordonner des données bruitées. Nous décrivons des liens avec d'autres problèmes combinatoires, en particulier pour des matrices modèlisant les données réelles d'ADN. Nous proposons des algorithmes adaptés, améliorant expérimentalement la robustesse sur données synthétiques et réelles, bien que moins clairement que la deuxième contribution.

La quatrième contribution présente le problème de sériation avec duplication, motivé par l'assemblage de génomes cancéreux via des données de conformation spatiale, que nous tentons de résoudre avec un algorithme de projections alternées fondé en partie sur les méthodes de sériation robuste, sur données synthétiques.

Contributions and thesis outline

Chapter 1: In this introductory chapter, we present the two problems that come into play in this thesis. First, we introduce the Seriation problem, its mathematical formulation, and develop two key methods to tackle this combinatorial problem over permutations, that will be employed further in the manuscript : the spectral relaxation, and convex relaxations. Then, we present the problem of de novo genome assembly, and explain the challenge caused by repeated regions (repeats ), before briefly describing the technology-specific data used in our experiments.

We conclude this chapter by stating the major challenges arising when trying to apply seriation to genome assembly.

Chapter 2: In this applicative chapter, we set up to use the spectral method to compute the layout of the reads, and integrate this layout module in a straightforward, end-to-end Overlap-Layout-Consensus assembly pipeline. We test the method against real, third generation sequencing DNA data of bacterial and yeast genomes. The proposed method is shown to be competitive, thus validating the use of seriation for genome assembly. However, due to repeats, the spectral method cannot layout all reads at once and eventually produces fragmented assemblies. For the bacterial data-sets, the assembled contigs can sometimes be merged into a single contig seamlessly, but the final yeast assemblies remain fragmented.

Chapter 3: This chapter presents a simple, yet powerful enhancement of the spectral method, drawing a parallel between Seriation and the Spectral Clustering method, well known in the machine learning community. We propose a unifying framework for seriation and circular seriation. In the circular variant of seriation, one seeks to find a circular (as opposed to linear) ordering of the elements based on their pairwise similarity. It has diverse applications in bioinformatics. Notably, finding the layout of circular genomes (such as the bacterial genomes used in Chapter 2) fits in the circular seriation framework. In this chapter, we bring together results from machine learning and specific matrix theory that shed light on the mechanisms underpinning the spectral method for seriation, and allow us to prove theoretical guarantees for circular seriation analogous to those existing for linear seriation in the noiseless case. Importantly, we designed an algorithm leveraging these results. It is a straightforward extension of the baseline spectral method, yet we show that it yields a valuable gain in robustness through numerical experiments. Remarkably, despite repeats, it correctly finds the layout of the bacterial genomes introduced in Chapter 2 in one shot (leading to a single contig).

Chapter 4: Here, we focus on the mathematical modeling of seriation through optimization problems, with the aim of finding algorithms that are by design more robust to the repeatinduced noise. We formalize the robust seriation problem and show that it is equivalent to a modified 2-SUM problem for a class of similarity matrices modeling those observed in DNA assembly. We explore several relaxations of this modified 2-SUM problem, and compare them empirically on synthetic data. The most salient and efficient methods are also evaluated on the bacterial genomes used in the previous chapters. One of them is able to correctly find the full layout of an E. coli genome in one shot from Oxford Nanopore reads. However, this method is experimentally more sensitive than the one presented in Chapter 3.

Chapter 5: This chapter introduces the problem of Seriation with Duplications. It is motivated by an application to cancer genome reconstruction, which is challenged by so-called structural variations. Namely, large portions of the genome, up to whole chromosomes, are duplicated or deleted, and new chromosomes are formed by fusing two pieces of chromosomes which are not connected in a normal genome. Hi-C (spatial conformation) data can be used to reconstruct the structure of such genome, but the duplications need to be addressed through a specific framework. After motivating and formalizing the problem of seriation with duplications, we propose an alternated minimization method, and evaluate it on synthetic data.

Chapter 6: This chapter concludes the thesis by summarizing our contributions, highlighting the key challenges addressed, and describing possible extensions and improvements of the present work.

Publications related to this manuscript are listed below.

• Chapter 2 is based on the following publication, Antoine Recanati, Thomas Brüls, and Alexandre d'Aspremont. A spectral algorithm for fast de novo layout of uncorrected long nanopore reads. Bioinformatics, 2016. The software is available on https://github. com/antrec/spectrassembler.

• Chapter 3 is based on the following report, Antoine Recanati, Thomas Kerdreux, and Alexandre d'Aspremont. Reconstructing latent orderings by spectral clustering. arXiv preprint arXiv:1807.07122,2 0 1 8 a . Ap y t h o np a c k a g ei sa v a i l a b l eo nhttps://github. com/antrec/mdso.

• Chapters 4 and 5 are based on the following report, Antoine Recanati, Nicolas Servant, Jean-Philippe Vert, and Alexandre d'Aspremont. Robust seriation and applications to Chapter 1

Introduction

Genome sequencing plays central role in biological research, with applications ranging from evolutionary science to human disease research. The process of whole genome sequencing, that is, reading the genome of a (member of a) species, involves two distinct tasks. First, the actual sequencing consists in collecting signal from a biological sample through a physical experiment (in the laboratory). Then, the assembly aims to reconstruct the genome from this signal, typically with dedicated algorithms (on a computer).

Over the past quarter-century, increase in computational power has facilitated genome sequencing through the collection and processing of larger amounts of data. The advent of new, "high throughput" sequencing technologies lead to an even more dramatic increase in sequencing power (and reduction in cost). Some of these "sequencing revolutions" have changed the game of genome assembly, calling for adapted methods on the algorithmic front.

Seriation is a mathematical problem akin to solving a one-dimensional jigsaw puzzle. One of the key steps of genome assembly essentially boils down to solving seriation. In this thesis, we will present algorithmic efforts to solve seriation, and investigate their efficiency when applied to genome assembly. Let us introduce the two core components of this thesis: seriation (theory and algorithms), and de novo genome assembly (application).

Seriation

In the following, we present the seriation problem, and introduce a few formal definitions and related problems. Then, we briefly review methods that have been proposed to solve it.

Presentation

The seriation problem seeks to recover a latent ordering from similarity information, such that similar elements are nearby in the final ordering. We typically observe a matrix measuring pairwise similarity between a set of n elements and assume they have a serial structure, i.e., they can be ordered along a chain where the similarity between elements decreases with their distance within this chain. In practice, we observe a random permutation of this similarity matrix, where the elements are not indexed according to that latent ordering. Seriation then seeks to find it back using only (local) pairwise similarity.

The problem was introduced in archaeology to find the chronological order of a set of graves [Robinson, 1951]. Each grave contained artifacts, assumed to be specific to a given time period. The number of common artifacts between two graves define their similarity, resulting in a chronological ordering where two contiguous graves belong to a same time period.

As a graphic illustration, let us consider the following example. The teapots dataset [Weinberger and Saul, 2006] is a collection of images of a rotating teapot, taken at angles regularly spaced between 0 and 360 . If the sequence of images is sorted by increasing angle, it constitutes a movie of the rotating teapot, making a full circle on itself. However, the collection of images is given unsorted. In order to recover the movie, we can compute the pairwise similarity between two images as the opposite of their `2 distance (the sum of the squared differences between the gray level in each pixel, if the image is black and white). Applying seriation to this set of similarities will output an ordering of the images where similar images are placed nearby, hopefully matching the ordering of the movie. Figure 1.1 shows the similarity matrix Similarity matrix between teapots images using the `2 distance, when the subscripts follow the ordering of the movie (1.1a), and when it is randomly permuted (1.1b).).

between images. Each entry (i, j) is the opposite of the sum of the squared pixel-wise distance.

In Figure 1.1a, the subscripts are ordered by increasing rotation angle (linearly spaced between 0 to 360 ). The similarity tends to decrease as we move away from the diagonal, i.e., when the difference of angles of the teapot images increases. In Figure 1.1b, the subscripts are given in a random order, which is what is observed in practice. We still observe maximal values on the main diagonal, which corresponds to self similarity between an image and itself. The diagonal of the matrix is invariant by permutation since the `2 distance between an image and itself is 0, for all images. The goal of seriation is to recover (1.1a) given (1.1b). As a qualitative result, The seriation problem has applications in DNA sequencing [Meidanis et al., 1998, Garriga et al., 2011] that we will develop throughout this manuscript. It also has applications in, e.g., envelope reduction [Barnard et al., 1995] and bioinformatics [Atkins and Middendorf, 1996, Higgs et al., 2006[START_REF] Cheema | Thread mapper studio: a novel, visual web server for the estimation of genetic linkage maps[END_REF], Jones et al., 2012] (see Liiv [2010] for a more thorough overview of applications).

Notations

Let us introduce notations in order to formulate the problem mathematically.

Matrices and vectors. S n is the set of real, symmetric matrices of dimension n,andS + n the set of non-negative, symmetric matrices of dimension n. The transpose of a matrix X is written X T , and we use the notation x T y for the dot product between two vectors x, y 2 R n (which can be seen as a matrix of size n⇥1), and sometimes also the standard notation hx, yi. 1 n =(1,...,1) T is the vector of size n with all ones. The sum of the entries of a vector x 2 R n can thus be written x T 1 n .F o ram a t r i xX of dimension n, diag(X) 2 R n is the vector constituting the main diagonal of X. However, if x 2 R n is a vector, diag(x) denotes the diagonal matrix whose main diagonal is x.W eu s ee k =(0,...,0, 1, 0,...,0) T for the k-th vector of the canonical basis of R n . I n is the identity matrix of dimension n. The sorted eigenvalues of a real, symmetric matrix X 2 S n are written 1 (X)  •••  n (X). We often omit the comma separating the two subscripts of a matrix, i.e.we may use both notations A i,j and A ij .

Permutations. The set of integers from 1 to n is written {1,...,n},o r[n] for short. Any permutation can be represented by a vector ⇡ 2 R n consisting in the rearrangement of the integers 1,...,n, where ⇡ i = j if and only if it moves the element at position j to position i. Such a permutation vector takes values in [n] and each value appears once in the vector. For ease of reading, we will use both notations ⇡ i and ⇡(i) to denote the i-th entry of the vector ⇡.

Alternatively, the same permutation can be represented by a permutation matrix Π 2 {0, 1} n⇥n such that Π ij =1if and only if ⇡ i = j. The two representations are equivalent and relate through the equation Πg = ⇡, where g =( 1 ,...,n) T denotes the identity permutation. The identity permutation matrix is the identity matrix I n . The matrix notation is convenient to write a matrix A whose entries have been permuted by ⇡. AΠ T is the matrix with coefficients A i,⇡(j) ,a n dΠAΠ T is the matrix whose entry (i, j) is A ⇡(i),⇡(j) . The set of permutations of n elements is written P n , and this notation can refer to the set of permutation vectors or matrices, depending on the context. Also, whenever the dimension is clear from the context, we may omit the subscript n in P n , 1 n , etc.

Mathematical Formulation and Related Problems

The main structural hypothesis on similarity matrices related to seriation is the concept of R-matrix, defined hereafter, using the terminology introduced in Atkins et al. [1998].

Definition 1.1.1. We say that the matrix A 2 S n is an R-matrix (or Robinson matrix) iff it is symmetric and satisfies A i,j  A i,j+1 and A i+1,j  A i,j in the lower triangle, where 1  j<i n.

These matrices are named after Robinson [1951]. The set of R matrices is written R in the following, and sometimes L R in Chapter 3 (to emphasize that the underlying structure is Linear, as opposed to Circular). Their entries are non increasing when moving away from the diagonal in a given row or column. An equivalent formulation is to say that, given any triplet 3 , with i  j  k,w eh a v eA ij A ik and A jk A ik .

(i, j, k) 2 [n]
We say that a symmetric matrix A is pre-R if there exists a permutation matrix Π such that the matrix ΠAΠ T (whose entry (i, j) is A ⇡(i),⇡(j) ) is an R-matrix. For such matrices, the seriation problem is to find a permutation that makes the matrix Robinson. Given a similarity matrix A that is pre-R, seriation can be written as a feasibility problem find Π 2 P such that ΠAΠ T 2 R.

(Seriation) Remark that the similarity matrix in Figure 1.1a is not an R-matrix. Indeed, although the similarity tends to decrease when moving away from the diagonal, it locally increases in some places. For instance, there are high similarity values between the last and the first images, given that the final orientation of the teapot is close to the initial. Yet, even when the strict (Seriation) problem is infeasible, we are still interested in finding an ordering such that (most) similar elements are placed nearby. This can be achieved by minimizing a well designed objective function. For instance, we can aim to minimize the number of anti-Robinson events, that is to say the number of violations of the two inequalities appearing in Definition 1.1.1.A discussion about such seriation criteria can be found in Hahsler [2017]. An objective function that will allow for spectral and convex relaxations in the following is the 2-SUM loss. The 2-SUM problem reads minimize

P n i,j=1 A ij |⇡ i ⇡ j | 2 such that ⇡ 2 P n .
(2-SUM)

Note that it is equivalent to minimize P n i,j=1 A ij |⇡ i ⇡ j | 2 and P n i,j=1 A i j |i j| 2 over the variable ⇡ 2 P n or 2 P n , since the optimal permutation of one of these problems is the inverse permutation of the other (⇡ ⇤ = 1 ⇤ ). Intuitively, the 2-SUM problem lays similar elements nearby as it penalizes the similarity between two elements by their squared distance in the ordering. It is also a particular case of the Quadratic Assignment Problem [Koopmans and Beckmann, 1957], written

min ⇡2Pn n X i,j=1
A i,j B ⇡(i),⇡(j) (QAP(A,B))

with B ij = |i j| 2 . Laurent and Seminaroti [2015] showed that for pre-R matrices A, Seriation is equivalent to QAP(A,B) when B 2 R n , i.e. when B has increasing values when moving away from the diagonal, and has constant values across a given diagonal (i.e., B is a Toeplitz matrix). This includes p-SUM problems, for p>0, corresponding to B ij = |i j| p . The case p =1is also known as the minimum linear arrangement problem (MLA) [George and Pothen, 1997]. For pre-R matrices, these problems are all equivalent and can be solved by a spectral algorithm in polynomial time, as we are about to see. However, when A is not pre-R, Seriation has multiple local solutions, and the spectral algorithm does not necessarily find a global optimum for 2-SUM,p -S U Mo rQAP(A,B) with B a Toeplitz, negated R matrix. In fact, these problems are NP-hard in general [Sahni and Gonzalez, 1976].

Optimization Strategies

We have seen that Seriation can be tackled by minimizing a loss function, such as the number of anti-Robinson events, the 2-SUM loss, or more generally some instances of QAP(A,B).Different methods can be used to perform the minimization.

Remark that the search space P n is discrete and of cardinality n!, thus preventing the use of exhaustive search (i.e., testing the values of the function over all possible permutations and pick the one with lowest score), even for small scale problems. To address this challenge, one may resort to relaxations of the problem. That is, replacing the hard, combinatorial problem with an easier, continuous one. By representing the permutation variable in a vector space such as R n , we can allow ("relax") the variable to take values in the space between some permutations, even though it does not represent a permutation anymore.

Spectral relaxations reformulate the problem into an eigen-problem, for which there exists efficient, polynomial time iterative algorithms. Convex relaxations let the variable take values in the convex hull of the initial set, meaning that instead of having a permutation variable ⇡ 2 P, we work with a variable x that can always be written x = P j ✓ j ⇡ (j) ,w i t h P j ✓ j =1 , ✓ j 0 and ⇡ (j) 2 P for all j. The loss function is also approximated by a convex function, if it is not already a convex function. Then, the arsenal of convex optimization, including first-order methods (such as gradient descent), can be used to solve the convex problem. These methods have convergence guarantees towards an optimal solution x ⇤ of the convex problem. However,

x ⇤ is most often not in the initial search space P, and its projection onto P may not be optimal for the initial problem.

Greedy Algorithms

Greedy methods typically solve sub-problems at a small neighborhood scale, with exhaustivesearch like procedures, and add up the bricks together to form the output sequence. For instance, branch and bound methods have been proposed for small scale seriation [Brusco, 2002]. However, they are impractical for problems of size larger than n ⇠ 100. Other heuristics from combinatorial optimization, e.g., simulated annealing and dynamic programming, have been proposed [Brusco et al., 2008], demonstrating very good experimental performance, but are still limited to small scale problems (n ⇠ 100) [ Evangelopoulos et al., 2017a, Hahsler et al., 2008].

Spectral Relaxation

The 2-SUM loss has been extensively used since it can be written as a quadratic, which is convenient for optimization. The following relaxation is also at the core of spectral clustering, where one seeks to cluster the data instead of ordering it (see the tutorial of Von Luxburg [2007] for details). It has a major importance in this thesis. It will serve as a baseline in the experiments, for it is scalable and efficient. Also, in Chapter 3, we will investigate extensions of this method.

For any real symmetric matrix A 2 S n , let D = diag(A1). D ij =0if j 6 = i,a n dD ii = P n j=1 A ij . D is called the degree matrix of A.I fA is the adjacency matrix of an undirected non-weighted graph, D ii is the degree of the node i. Now, let L = D A be the Laplacian of A. For any vector f 2 R n ,w eh a v e

f T Lf = 1 2 n X i,j=1 A ij (f i f j ) 2 . (1.1)
Indeed, observe that

f T Lf = f T Df f T Af = P n i=1 f 2 i D ii P n i,j=1 A ij f i f j = P n i=1 f 2 i ( P n j=1 A ij ) P n i,j=1 A ij f i f j = P n i,j=1 A ij (f 2 i f i f j ) = 1 2 P n i,j=1 A ij (f 2 j + f 2 i 2f i f j ) = 1 2 P n i,j=1 A ij (f i f j ) 2 , (1.2)
where we have used the symmetry of A on the penultimate line, and the fact that the symbols of the subscripts i and j could be switched. Hence, 2-SUM can be written as a quadratic optimization problem on permutation vectors, minimize ⇡ T L⇡ such that ⇡ 2 P n .

(2-SUM (quad.))

The spectral relaxation relies on the analysis of the spectrum of the Laplacian. Without loss of generality, we will consider in the following that the similarity matrices have non-negative entries. Remark that adding an offset to a matrix A to make it non-negative does not change the optimal permutation in 2-SUM, since it translates into an offset in the objective function that is independent of the permutation. From Equation (1.1), we can see that when L is the laplacian of a (symmetric, non-negative) similarity matrix A, it has non-negative eigenvalues.

Indeed, recall that the eigenvector associated to the smallest eigenvalue of a matrix M 2 S n is given by

f 1 2 argmin kf k 2 =1 f T Mf, (1.3)
and for i>1,t h ei-th smallest eigenvector is given by

f i 2 argmin kf k 2 =1,f T f j =0,j<i f T Mf, (1.4)
Now, observe that the right-hand-side of Equation (1.1) is always non-negative, hence no eigenvector f can have negative eigenvalue. Note also that 1 is an eigenvector of L, with associated eigenvalue 0. Indeed, if all f i in Equation The membership to P can be equivalently enforced by the three previous constraints. The spectral relaxation of 2-SUM (quad.) studied in Atkins et al. [1998] basically drops the (integer constraint), as noted in [START_REF] Ding | Linearized cluster assignment via spectral ordering[END_REF]. Since 1 is in the nullspace of L, the objective in 2-SUM (quad.)

does not vary by subtracting ((n + 1)/2)1 to the permutation vectors. Hence, the (sum constraint) can be transformed into ⇡ T 1 =0in the 2-SUM problem. Finally, the quadratic 2-SUM objective is homogeneous, and we can therefore chose to rescale the permutation vectors and transform the (norm constraint)i n t ok⇡k 2 2 =1 . All in all, the spectral relaxation of 2-SUM reads, minimize f T Lf such that f T 1 =0, kf k 2 =1.

(Spectral Relax.)

Given that 1 is the first eigenvector of L, we recognize Equation (1.4) with i =2 . The solution f ⇤ of the Spectral Relax. is the eigenvector associated to the second smallest eigenvalue of the laplacian of A. However, since we relaxed the integer constraint,t h es o l u t i o nf ⇤ is in general not a permutation. In order to recover a permutation from f ⇤ , we can project f ⇤ back onto the set of permutation vectors, i.e.,fi n d⇡ 2 argmin ⇡2Pn kf ⇤ ⇡k 2 . The computation of this projection actually boils down to sorting the entries of f ⇤ : ⇡ is such that f ⇤ (⇡(1))  ...  f ⇤ (⇡(n)). Note that this projection ⇡ ⇤ is not guaranteed to be the optimum of 2-SUM in general. If A is the adjacency matrix of a disconnected graph, i.e., there are connected components in the graph with no edges in between, then f ⇤ has constant values inside each connected component (and can be used for clustering [START_REF] Von | A tutorial on spectral clustering[END_REF]). This makes the projection degenerate, since there is no best way to sort a constant vector. However, Seriation and 2-SUM aim to find a global ordering integrating all the local similarities. If the graph is disconnected, say, in two clusters, then only the sub-orderings restricted to each cluster will matter. Therefore, we restrict ourselves to the case where A is the adjacency matrix of a connected graph (for any pair of nodes (i, j), there exists a path going from i to j). Then, the second smallest eigenvalue, called the Fiedler value, is positive : 2 > 0, and there are meaningful variations in f ⇤ . Some values of f ⇤ can still be equal, and Atkins et al. [1998] propose a method to deal with these degeneracies with so-called PQ-trees, but this situation is scarcely encountered in practice when dealing with real or noisy data.

We summarize the spectral method in Algorithm 1.1. A major result from Atkins et al.

Algorithm 1.1 Spectral ordering [Atkins et al., 1998] Input: Connected similarity matrix A 2 R n⇥n 1: Compute Laplacian L A = diag(A1) A 2: Compute second smallest eigenvector of L A , f 1 3: Sort the values of f 1 Output: Permutation : f 1 ( (1))  ...  f 1 ( (n)) [1998] states that Algorithm 1.1 solves Seriation whenever it is feasible. This is formalized in

Convex Relaxations

The spectral relaxation is powerful, but it is intrinsically specific to the (2-SUM (quad.)) objective, and may perform poorly when the input matrix A is noisy. Convex relaxations are more flexible and come with theoretical guarantees, though they are often heavier computationally.

Formally, consider the constrained optimization problem, minimize f (x)

subject to x 2 C, (1.5) in x 2 R d
, where f is a smooth (its gradient is Lipschitz continuous) convex function (e.g.,t h e (2-SUM) objective) and C is a closed set (e.g., P). The convex relaxation approach approximates the non-convex set C with a convex one. Specifically, it lets the variable be in a convex set that contains the original set C. The tightest relaxation consists in considering the convex hull of the original set. For the record, the convex hull of a set C, denoted hull(C), is the set of all convex combinations of points in C,

hull(C)= ✓ 1 x 1 + ...+ ✓ k x k | ✓ i 0,i=1,...,k, ✓ 1 + ...+ ✓ k =1 .
It is the smallest convex set that contains C. The convex relaxation of (1.5) reads minimize f (x)

subject to x 2 hull(C). (1.6)
Then, convex optimization methods can be used to solve (1.6). For instance, projected gradient descent is a first order iterative method. Given the current iterate x t , it computes the gradient of the function at x t , takes a step in the opposite direction of the gradient, and projects back onto the constraint set (since x t+1/2 may no longer be in hull(C)), with,

x t+1/2 = x t rf (x t ), (gradient step)

x t+1 = argmin x2hull(C) kx x t+1/2 k. (projection step)
In practice, although we can mathematically define hull(C), it may be computationally challenging to perform the projection step.

Permutation matrices

The set of permutation matrices can be written with the following constraints,

P n = n Π 2 {0, 1} n⇥n Π1 = 1, Π T 1 = 1 o . (1.9)
The two stochastic constraints impose that in a permutation of the integers 1,.

B n = {X 2 R n⇥n | X 0,X1 = 1,X T 1 = 1}.
It is a polyhedron, i.e., it is defined by linear constraints. Work akin to seriation involving B include that of Vogelstein et al. [2011], who used the conditional gradient Algorithm 1.2 to minimize the objective of QAP(A,B) over the Birkhoff polytope B.

Each iteration of the algorithm involves solving a linear program in B (1.7), which is achieved using a Hungarian matching algorithm [Kuhn, 1955]. Again, the membership to B needs not to be enforced explicitly when using the conditional gradient algorithm, since the sequence of iterates remain in B by construction (using convex combinations of points in B). which makes this approach limited to medium-scale problem. Refinements of this approach, using for instance the Frank-Wolfe algorithm, can make the problem scale to larger sizes [START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF].

Permutation vectors

A permutation vector has only n entries (compared to n 2 for a permutation matrix). Therefore, a relaxation on the set of permutation vectors may be more scalable. We have seen earlier three constraints defining the set of permutation vectors (also written P in this subsection). Still, the (norm constraint) is not linear. Yet, convex optimization routinely solves problems with a quadratic objective and linear equality and inequality constraint, but quadratic constraints are often challenging, hence this formulation is not adapted to convex optimization. The set of permutation vectors can also be defined as follows [START_REF] Han | Beyond the birkhoff polytope: Convex relaxations for vector permutation problems[END_REF],

⇡(i) 2 [n],i =1,...,n (integer constraint) ⇡ T 1 = n(n + 1)/2 (sum constraint) X i2S ⇡ i  |S| X i=1 (n +1 i) for all S ⇢ [n] (partial sum constraints)
The convex hull of the set of permutation vectors is also obtained by relaxing the (integer constraint), and is defined as follows In a nutshell, the extended formulation states that a vector x belongs to PH n if it constitutes the first n entries of a larger vector respecting some linear inequality constraints,

PH n = 8 < : x 2 R n n X i=1 x i = n(n + 1) 2 , X i2S x i  |S| X i=1 (n +1 i) for all S ⇢ [n] 9
x 2 PH n if x 2 {x in | (x in ,x rest ) 2 SN n }, (1.11)
where SN n is a polyhedron, i.e., the membership of x extd. The key observation is that the LMO (1.7) can be computed efficiently here, as it boils down to sorting the entries of the gradient (which is a vector of size n). Indeed, if y denotes the gradient of f at x t ,E q u a t i o n( 1.7) can be written as, minimize P n i=1 y i s i subject to s 2 PH n .

(1.13) Now, observe that the minimum of a linear function on a polyhedron resides on a vertex, hence we can restrict the variable s to be in P n instead of PH n . The permutation that minimizes (1.13) is the one with maximal weight (n) on the largest entry of y, with second maximal weight (n 1) on the second largest entry of y, etc. Formally, if ⇡ is the permutation that sorts the entries of s increasingly,

y ⇡(1)  y ⇡(2)  ...  y ⇡(n)
the solution s ⇤ of Equation (1.13) is the inverse permutation of ⇡, defined by,

s ⇤ ⇡(k) = k, for k =1,...,n.
Algorithm 1.2 can be implemented with hull(C)=PH n , where the LMO (1.7) consists in sorting the entries of the gradient as described previously, which has algorithmic complexity

O(n log n).

Symmetry Breaking

A crucial issue with the convex relaxations presented above is that the optimum of the convex problems (1.10)a n d( 1.12) are trivial and non-informative.

Indeed, recall that 1 is in the nullspace of L,a n dt h a tL is positive semi-definite. The geometrical center of PH n , c n = n+1 2 1 n , therefore minimizes (1.12) (where the objective is 0). Similarly, the geometrical center of B n , C n = 1 n 1 n 1 T n , minimizes (1.10). Yet, the respective centers of PH n and B n are at the same distance from any permutation (vector or matrix, respectively). Therefore, the task of projecting them back onto the set of permutations is totally degenerate. This is essentially due to the following symmetry. The 2-SUM objective is invariant by flipping a permutation. For instance, (1,...,n) and (n, . . . , 1) yield equal score. Formally, the operator T n defined by T n (⇡)=(n + 1) ⇡ leaves 2-SUM invariant. In order to overcome this issue, [START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF] augmented the convex relaxation (1.10) as follows.

-Introduce a tie-breaking constraint, ⇡ 1 +1 ⇡ n , or in matrix form, e T 1 Πg +1 e T n Πg, to resolve ambiguity about the direction of the ordering.

-Add a penalty to the Froebenius norm of P Π, with

P = I n C n = I n 1 n 1 n 1 T
n ,t op u s h the solution away from the center C n .

Additionally, their convex formulation allows to incorporate ordering constraints of the form ⇡ i ⇡ j  k , to leverage prior knowledge on the ordering (and, if this prior knowledge is consistent, help breaking the symmetry). Also, they average over several perturbations of g to gain in robustness. All in all, the enhanced problem (1.10) reads,

minimize 1 p Tr(Y T Π T L A ΠY ) µ p kP Πk 2 F subject to DΠg  , Π1 = 1, Π T 1 = 1, Π 0, (Matrix-Relax. 2-SUM)
where DΠg  contains the tie-breaking and aprioriconstraints, the second line of constraints

imposes Π 2 B n , each column of Y 2 R n⇥p is a perturbed version of g,andµ is a regularization parameter. Keeping µ< 2 (L A ) 1 (YY T ) ensures that (Matrix-Relax. 2-SUM) remains convex.
Lim and Wright [2014] adapted these improvements to the permutation vector formulation, yielding the following problem,

minimize x T Lx µkPxk 2 2 subject to Dx  , x 2 PH n . (Vector-Relax. 2-SUM)
The objective of (Vector-Relax. 2-SUM) can also be written x T (L µP )x, making it clear that the objective remains convex when the regularization parameter is smaller than the Fiedler value, i.e., µ< 2 (L). However, rather than choosing a fixed value of the regularization parameter µ, they iteratively increase it in outer-loops of a continuation (a.k.a, graduated non-convexity) scheme. They start with µ< 2 (L), and increase it until µ> n (L). Their approach produces a sequence of solutions to sub-problems following a path from c n to a permutation (i.e., a vertex of PH n ).

Applications to Genomics

Let us present the outline of de novo genome assembly, one of the key challenges (the repeats ), and some specific sequencing technologies. However, these fragments are given "in a bag", i.e., we ignore their positions and their order in the genome (they are correctly located in the Figure for illustration only). Thanks to the overlaps, we can recover the full sequence from the fragments.

De novo Genome Assembly

In practice, we use a device that outputs an electronic signal from a DNA fragment. The sequence can then be deduced from the signal (this inference process is called basecalling).

However, due to limitations inherent to the physicochemical process enabling us to "read" the DNA sequence, we can only access partial sub-fragments (called reads hereafter) extracted from the input DNA strand.

To overcome this limitation, the idea of shotgun sequencing is to clone the genome multiple times, and sequence pieces of the clones at random locations in the genome. Consequently, the genome is oversampled and all parts are covered by multiple reads with high probability. Hence, we have redundant information : there are overlaps between the reads, and we can assemble all the pieces together to retrieve the input DNA strand. For instance, the example sequence 'AATCGCA' could yield the three following sub-sequences in an experiment, {'AATC','ATCG','CGCA'}. We see that 'AATC' overlaps 'ATCG' and 'ATCG' overlaps 'CGCA', enabling us to reconstruct the full word. Nonetheless, we ignore the locations of the reads on the genome, and we must infer them from the overlap information, in the way of a jigsaw puzzle. A schematic illustration is given in Figure 1.4.

In some applications, we wish to sequence the genome of the member of a species for which we already have a reference genome (for instance, a human). Then, we can find the locations of the reads by mapping them to a reference before reconstructing the full sequence of the given individual. In contrast, de novo genome assembly refers to the task of reconstructing the whole DNA strand from the fragments (reads), sampled at random locations, without any reference.

In a de novo assembly experiment, the coverage c is the sum of the lengths of the reads divided by the length of the genome. In average, if all reads had the same length and were sampled uniformly along the genome, a given read would overlap with c other reads.

The DNA has a double-strand structure, as shown on Figure 1.5.E a c h s t r a n d h a s a n orientation determined by the direction in which DNA was replicated (from one end denoted 5' to the other, denoted 3'). The two strands are complementary, i.e., one is the reverse complement of the other, where the reverse complement of a sequence (s 1 ,...,s n ) is defined as (s n ,...,s 1 ), with Ā = T , T = A, C = G,and Ḡ = C. In an assembly experiment, the reads may come from either strand. Therefore, it is necessary to consider both possible orientations when computing the overlaps. Also, when using the Seriation framework, the orientation information is not taken into account in the similarity matrix, but the layout needs to be consistent with the orientation constraints (one read has only one global orientation). Overlap-Layout-Consensus (OLC) is a major assembly paradigm based on three main steps.

First, compute the overlaps between all pairs of read. This provides a similarity matrix A, whose entry (i, j) measures how much reads i and j overlap (and is zero if they do not).

Then, determine the layout from the overlap information, that is to say find an ordering and positioning of the reads that is consistent with the overlap constraints. Finally, given the tiling of the reads obtained in the layout stage, the consensus step aims at determining the most likely DNA sequence that can be explained by this tiling.

Computing the overlaps between all pairs of reads involves n(n 1)/2 pairwise comparisons.

Dynamic programming can be used to perform sequence alignment [Smith and Waterman, 1981] and provide an overlap score. However, such methods are impractical with n ⇠ 10 4 reads.

Instead, methods based on hashing [START_REF] Li | Minimap and miniasm: fast mapping and de novo assembly for noisy long sequences[END_REF], Berlin et al., 2015] can be used. In this work, we will use such software to compute the overlaps (sometimes called overlapper) as a black-box, providing, for all pairs of sequences, 1. an overlap score (length of the overlap, if any) and 2. overlap detailed information, i.e., position of the overlap on each of the two sequences, and mutual orientation (a DNA strand can be read in two possible directions, as shown in Figure 1.5, and the orientation of an overlap indicates whether the two reads come from the same strand or from opposite strands).

The layout step, akin to solving a one dimensional jigsaw puzzle, is a key step in the assembly process, and fits in the framework of Seriation. In an ideal setting, a given read has a significant overlap with the next read, a smaller overlap with the one after, and so on, until it no longer overlaps the subsequent reads. For instance, on Figure 1.4, the leftmost read (darkblue) has a large overlap with the second read (blue), a small overlap with the third one (light blue), and does not overlap the following reads. Hence, the similarity matrix from an ideal genome assembly experiment is an R-matrix. Also, it is a sparse, banded matrix (it has non-zero values only within a band, corresponding to the maximal distance between two overlapping reads).

Finally, the consensus can be performed through multi-sequence alignment. In the above example with the 'AATCGCA' sequence, it could be recovered from the three reads as follows, AATC AATCG AATCGCA AATCGCA However, in practice, there are basecalling errors in the reads. There can be substitution errors (a nucleotide is replaced by another one), insertions (a nucleotide is added in the sequence), and deletions (a nucleotide is removed from the sequence). Performing the consensus therefore requires more than majority-vote like rules. Hopefully, efficient implementations of algorithms based on dynamic programming exist for multiple sequence alignment, with reasonable numbers and sizes of sequences to align [Lee et al., 2002[START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF].

Repeated Regions (repeats)

The basecalling errors represent a challenge in the assembly process, as they may induce errors in the overlap computation (leading to erroneous layout), and make the consensus derivation more challenging. Yet an even more challenging issue is the presence of repeated regions (called repeats ), i.e., stretches of DNA that occur multiple times in near-identical copies throughout the genome. Repeats can be as long as a few thousands of nucleotides. As illustrated in

R A R B R r 1 a 1 a 2 r 2 r 3 b 1 b 2 r 4 r 1 b 1 b 2 r 2 r 3 a 1 a 2 r 4 correct mis-assembled

Sequencing technologies

Several sequencing technologies exist and produce data with different characteristics, leading to technology-specific algorithmic paradigms. The most widely used sequencing technologies fall into the following categories (see [START_REF] Nagarajan | Sequence assembly demystified[END_REF] for a more complete survey of sequencing tools and assembly algorithms).

Next Generation Sequencing (NGS)

What was formerly called Next Generation Sequencing (NGS) or High Throughout sequencing usually refers to several short-reads technologies that parallelize the sequencing process. One of the most widely used is commercialized by Illumina and based on the sequencing by synthesis process. It synthesizes and amplifies the reads with DNA polymerase, and uses imagery with fluorescent markers to read the (short) sub-sequences base per base. These technologies have dramatically reduced the cost of DNA sequencing in the late 1990s.

Typical data produced with Illumina are millions of reads of a few hundreds bases, with accuracy exceeding 99% (the read accuracy is the proportion of correctly sequenced bases, as opposed to the sequencing errors listed above [substitution, deletion, insertion]). Such sequencers provide additional information to be used in the assembly: the reads are given in pairs, and we know the distance between two paired reads (it is the same for all pairs and is substantially larger than the reads length). This adds structural constraints on the layout, and provides information of longer range than the overlaps.

Third Generation Sequencing

More recently, modifications to the sequencing by synthesis technology by Pacific Biosciences (PacBio) gave rise to Single Molecule Real Time Sequencing (SMRT), a method capable of producing reads tens of thousands nucleotides (10 4 b) long. However, the reads accuracy dropped to ⇠ 87%.

An even more recent long-reads technology, introduced by Oxford Nanopore Technology (ONT), is based on the observation of ion current when a DNA strand passes through a nanopore. It also produces reads of a few tens of thousands bases with lower accuracy than short-reads, although the reads accuracy tends to improve and is claimed to reach 92 to 97% now.

Despite its lower accuracy, such "long-reads" technology is highly valuable for de novo assembly, since the length of the reads is larger than most repeats, making them easier to resolve.

Moreover, while short-reads assemblers have been enhanced with number of handcrafted heuristics throughout the years, long-reads assemblers are quite novel and the algorithmic design of dedicated assemblers is still burgeoning, as the technology keeps evolving. Therefore, the de novo assembly experiments conducted throughout this thesis will use long-reads.

State of the Art of Assembly Methods

Let us briefly review the key principles used in assembly computational tools.

Assembly paradigms

Most assembly methods rely on (at least) one of the following paradigms: greedy methods, De Then, the layout step searches for an ordering and positioning of the reads consistent with the overlap information. Notably, overlapping reads must be placed nearby in the layout found.

For instance, a method akin to De Bruijn graphs could be applied to the layout step, where the nodes of the graph of interest are the reads, and edges represent the overlap (if any) between the reads. Then, the layout can be found by searching for a path that goes through all nodes once, where the sum of the weights along the path is maximized (Hamiltonian path problem).

Finally, once the layout of the reads is obtained, the consensus step derives a sequence in a majority-vote fashion, with, e.g., multiple sequence alignment.

Assembly pipelines

In practice, an approach such as DBG fails to uniquely assemble genomes in the presence of repeats. Hence, in order to meet the challenges caused by repeats and sequencing errors, assembly software consist in pipelines involving several components. Most NGS assembly pipelines involve the three following steps : 1. Contig generation; 2. scaffolding,3 . finishing.

The contigs are contiguous genomic fragments, i.e., the result of the assembly of a subsequence of the whole genome. Typically, heuristics are used to identify reads originating from repeats, and the overlap graph is cut into separate, repeat-free connected components. In these repeat-free sub-graphs, a De Bruijn graph approach can be successfully applied to perform the partial assembly of the contigs. Thus, this first step assembles the "easy" regions with no ambiguities.

Then, the scaffolding consists in determining the layout of the contigs (which can be thought of as blocks of pieces of jigsaw puzzle, where a single piece would be a read) together, i.e., Although there is no mate pair or pair end side information with third generation data to be used in the scaffolding, the length of the reads allow to resolve contigs smaller than the read length, resulting in larger contigs than with NGS. In short, if we split the full human genome into n bins (of typical length 10 4 10 6 basepairs each), an Hi-C experiment produces an n ⇥ n interaction matrix A such that A ij is the frequency of interactions between DNA fragments in bins i and j. It roughly proceeds as follows. First, freeze the DNA in its current 3D conformation, and collect pairs of DNA fragments that lie close to each other in this spatial conformation, thanks to a ligation process. For every such pair (k, l), each of the two fragments is then mapped to a reference genome, providing their positions, p k and p l . Finally, add +1 to the interaction matrix entry A ij corresponding to the two bins i and j that respectively span p k and p l . This process is repeated to statistically obtain an average proximity (frequency) between two bins.

determining
Interestingly, the frequency of 3D interactions tends to decrease with the distance between the fragments. Thus, the layout of the bins can be obtained by applying Seriation to Hi-C data (although this is not the original purpose of Hi-C data). The GRAAL assembler [Marie-Nelly et al., 2014] uses a probabilistic model of the interaction to compute the most likely genome structure from the contact maps.

10X Genomics

A recent development in sequencing technology commercialized by 10X genomics combines short-reads with so-called molecular barcoding, linking short-reads to long molecules to provide long-range information. In short, a barcode is associated to regions of DNA of large length (a few tens of kbp), and two reads that are close to each other on a DNA strand are likely to share several barcodes. This barcoding notably permits to call structural variants and distinguish between haplotypes for diploid genomes (such as the human genome). The Supernova assembler [Weisenfeld et al., 2017] is based on a short-reads assembly scheme, and the additional molecular barcode information is used to disambiguate the scaffolding.

Challenges

We have introduced a mathematical problem, seriation, and an applicative problem, de novo genome assembly. Although with idealistic data, the latter problem would fit seamlessly as an instance of the first, we have seen that in practice, it does not. For instance, applying the Spectral Algorithm 1.1 to a similarity matrix constructed from a real sequencing experiment yields a corrupted ordering, as one can see in Figure 1.7b. Yet, Theorem 1.2.1 guarantees that Algorithm 1.1 solves Seriation. Therefore, this experiment does not fit "as is" in the framework of Seriation. Let us highlight three key challenges when trying to apply seriation to de novo genome assembly.

• Robustness. As observed in Figure 1.7, the repeats induce a number of out-of-diagonal points on the similarity matrix (1.7a), which can be decomposed as the sum of a banded (and, in theory, Robinsonian) matrix, and a sparse noise matrix out of the band. This sparse noise suffices to make the spectral Algorithm 1.1 fail, although it has theoretical guarantees in the noiseless case. One of the key challenges is to design algorithmic schemes that are robust to variations from a noiseless R-matrix to a noisy observation of it. In fact, the repeats induce a specific kind of noise. Ideally we would like to be robust to it.

• Scalability. Typical de novo genome experiments with long reads involve similarity matrices of size n ⇠ 10 4 . Many seriation algorithms are impractical at such a scale. Hence, a major challenge is to design algorithms for seriation that are scalable.

• Dealing with multiple strands. Another challenge that we have not mentioned so far is that many species are eukaryotes (have cells with a nucleus) and their genome has multiple chromosomes, i.e., it is composed of several DNA strands. As a result, we do not wish to find one sequence, but several sequences (one per chromosome). However, there are overlaps between reads sampled from distinct chromosomes. Hence, the clustering-inchromosomes step is non trivial, and cannot be done easily as a pre-processing, to break down the problem into several single-strand assembly problems.

Chapter 2

Application of the Spectral Method to Genome Assembly

This chapter presents a direct application of seriation to de novo assembly. From a mathematical perspective, it solely relies on the material introduced in 1.2.2. Specifically, the spectral Algorithm 1.1 is employed.

The main goal here was to get our hands dirty with real sequencing data, and make the proof of concept that seriation is an adequate framework for de novo assembly. To this end, we developed a pipeline that takes DNA reads (in fasta or fastq format) as input and provides an assembled sequence (or a set of sequences) in output. The overlaps between pairs of reads are computed with standard software. Algorithm 1.1 is applied to a similarity matrix built from the overlaps in a straightforward fashion. Then, given the layout found by seriation, a consensus sequence is derived via multiple sequence alignment, with dedicated software.

Despite the repeats issue presented in 1.3.2, this simple pipeline yields perhaps surprisingly good results. The challenge posed by repeats is addressed with the so-called bandwidth heuristic, which is a quick and dirty way to tackle the shortcomings of the spectral method. The underlying observation is that long overlaps are more likely to be true rather than repeat-induced overlaps. This seems to hold thanks to the length of the reads used here (third generation sequencing).

In the couple of years that followed this work, some related methods have emerged or evolved. For instance, we used the minimap [START_REF] Li | Minimap and miniasm: fast mapping and de novo assembly for noisy long sequences[END_REF] tool to compute overlaps between reads, and GraphMap [START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF] to align the reads to a reference sequence in order to plot the layout versus a reference. Now, the minimap2 tool [Li, 2018] has been released and can perform both tasks. Also, the reads accuracy may have slightly improved, and the Oxford Nanopore preset modes of standard assemblers such as canu [Koren et al., 2017] incorporate more dataspecific refinements. Still, the related work for de novo assembly of long reads outlined in the introduction of this chapter remains broadly accurate.

The content of this chapter is based on the following publication, Antoine Recanati, Thomas Brüls, and Alexandre d'Aspremont. A spectral algorithm for fast de novo layout of uncorrected long nanopore reads. Bioinformatics,2 0 1 6 .

Supplementary material for this chapter is given in Appendix Chapter A.

Chapter Abstract

Motivation: New long read sequencers promise to transform sequencing and genome assembly by producing reads tens of kilobases long. However, their high error rate significantly complicates assembly and requires expensive correction steps to layout the reads using standard assembly engines.

Results:

We present an original and efficient spectral algorithm to layout the uncorrected nanopore reads, and its seamless integration into a straightforward overlap/layout/consensus (OLC) assembly scheme. The method is shown to assemble Oxford Nanopore reads from several bacterial genomes into good quality (⇠99% identity to the reference) genome-sized contigs, while yielding more fragmented assemblies from the eukaryotic microbe Sacharomyces cerevisiae.

Availability and implementation: https://github.com/antrec/spectrassembler.

Introduction

De novo whole genome sequencing seeks to reconstruct an entire genome from randomly sampled sub-fragments whose order and orientation within the genome are unknown. The genome is oversampled so that all parts are covered multiple times with high probability.

High-throughput sequencing technologies such as Illumina substantially reduced sequencing cost at the expense of read length, which is typically a few hundred base pairs long (bp) at

best. Yet, de novo assembly is challenged by short reads, as genomes contain repeated sequences resulting in layout degeneracies when read length is shorter or of the same order than repeat length [Pop, 2004].

Recent long read sequencing technologies such as PacBio's SMRT and Oxford Nanopore Technology (ONT) have spurred a renaissance in de novo assembly as they produce reads over 10kbp long [Koren and Phillippy, 2015]. However, their high error rate (⇠15%) makes the task of assembly difficult, requiring complex and computationally intensive pipelines.

Most approaches for long read assembly address this problem by correcting the reads prior to performing the assembly, while a few others integrate the correction with the overlap detection phase, as in the latest version of the Canu pipeline [Koren et al., 2017] (former Celera Assembler [Myers et al., 2000]).

Hybrid techniques combine short and long read technologies: the accurate short reads are mapped onto the long reads, enabling a consensus sequence to be derived for each long read and thus providing low-error long reads (see for example [START_REF] Madoui | Genome assembly using nanopore-guided long and error-free dna reads[END_REF]). This method was shown to successfully assemble prokaryotic and eukaryotic genomes with PacBio [Koren et al., 2012]a n dO N T [ Goodwin et al., 2015]d a t a . Hierarchical assembly follows the same mapping and consensus principle but resorts to long read data only, the rationale being that the consensus sequence derived from all erroneous long reads matching a given position of the genome should be accurate provided there is sufficient coverage and sequencing errors are reasonably randomly distributed: for a given base position on the genome, if 8 out of 50 reads are wrong, the majority vote still yields the correct base. Hierarchical methods map long reads against each other and derive, for each read, a consensus sequence based on all the reads that overlap it. Such an approach was implemented in HGAP [START_REF] Chin | Nonhybrid, finished microbial genome assemblies from long-read smrt sequencing data[END_REF] to assemble PacBio SMRT data, and more recently by Loman et al. [2015], to achieve complete de novo assembly of Escherichia coli with ONT data exclusively.

Recently, [START_REF] Li | Minimap and miniasm: fast mapping and de novo assembly for noisy long sequences[END_REF] showed that it is possible to efficiently perform de novo assembly of noisy long reads in only two steps, without any dedicated correction procedure: all-vs-all raw read mapping (with minimap) and assembly (with miniasm). The miniasm assembler is inspired by the Celera Assembler and produces unitigs through the construction of an assembly graph. Its main limitation is that it produces a draft whose error rate is of the same order as the raw reads.

Here, we present a new method for computing the layout of raw nanopore reads, resulting in a simple and computationally efficient protocol for assembly. It takes as input the all-vs-all overlap information (e.g. from minimap, MHAP [Berlin et al., 2015] or DALIGNER [START_REF] Myers | Efficient local alignment discovery amongst noisy long reads[END_REF]) and outputs a layout of the reads (i.e., their position and orientation in the genome).

Like miniasm, we compute an assembly from the all-vs-all raw read mapping, but achieve improved quality through a coverage-based consensus generation process, as in nanocorrect [Loman et al., 2015], although reads are not corrected individually in our case.

The method relies on a simple spectral algorithm akin to Google's PageRank [START_REF] Page | The pagerank citation ranking: Bringing order to the web[END_REF] with deep theoretical underpinnings, described in §2.2.1. It has successfully been applied to consecutive-ones problems arising in physical mapping of genomes [Atkins and Middendorf, 1996], ancestral genome reconstructions [Jones et al., 2012], or the locus ordering problem [START_REF] Cheema | Thread mapper studio: a novel, visual web server for the estimation of genetic linkage maps[END_REF], but to our knowledge has not been applied to de novo assembly problems.

In §2.2.2, we describe an assembler based on this layout method, to which we add a consensus generation step based on POA [Lee et al., 2002], a multi-sequence alignment engine. Finally, we evaluate this pipeline on prokaryotic and eukaryotic genomes in §3.4, and discuss possible improvements and limitations in §2.4.

Methods

Layout computation

We lay out the reads in two steps. We first sort them by position, i.e., find a permutation ⇡ such that read ⇡(1) will be positioned before read ⇡(2) on the genome. Then, we iteratively assign an exact position (i.e., leftmost basepair coordinate on the genome) to each read by using the previous read's position and the overlap information.

The key step is the first one, which we cast as a seriation problem, i.e., we seek to reconstruct a linear order between n elements using unsorted, pairwise similarity information [Atkins et al., 1998[START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF]. Here the n elements are the reads, and the similarity information comes from the overlapper (e.g. from minimap).

The seriation problem and the spectral relaxation have been discussed in the introductory Chapter 1. For self-containment, we briefly recall the formulation leading to the spectral relaxation. Given a pairwise similarity matrix A ij , and assuming the data has a serial structure, i.e.that there exists an order ⇡ such that A ⇡(i)⇡(j) decreases with |i j|, seriation seeks to recover this ordering ⇡ (see Figure 1.3 in Chapter 1, repeated here in Figure 2.1 for an illustration). If such an order ⇡ exists, it minimizes the 2-SUM score,

2-SUM(⇡)= n X i,j=1 A ij ⇡(i) ⇡(j) 2 , (2.1)
and the seriation problem can be solved as a minimization over the set of permutation vectors [START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF]. In other words, the permutation ⇡ should be such that if A ij is high (meaning that i and j have a high similarity), then ⇡(i) ⇡(j) 2 should be low, meaning that the positions ⇡(i) and ⇡(j) should be close to each other. Conversely, if A ij =0,t h ep o s i t i o n s of i and j in the new order may be far away without affecting the score. When using seriation to solve genome assembly problems, the similarity A ij measures the overlap between reads i and j. In an ideal setting with constant read length and no repeated regions, two overlapping reads should have nearby positions on the genome. We therefore expect the order found by seriation to roughly match the sorting of the positions of the reads.

The problem of finding a permutation over n elements is combinatorial. Still, provided the original data has a serial structure, an exact solution to seriation exists in the noiseless case [Atkins et al., 1998] using spectral clustering, and there exist several convex relaxations allowing explicit constraints on the solution [START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF].

The exact solution is directly related to the well-known spectral clustering algorithm. Indeed, for any vector x, the objective in (2.1)r e a d s

n X i,j=1 A ij x i x j 2 = x T L A x, L A = diag(A1) A
where L A is the Laplacian matrix of A. This means that the 2-SUM problem amounts to

min ⇡ ⇡ T L A ⇡
where ⇡ is a permutation vector. Roughly speaking, the spectral clustering approach to seriation relaxes the constraint "⇡ is a permutation vector" into "⇡ is a vector of R n orthogonal to the constant vector 1 =( 1 ,...,1) T " with fixed norm. As we have seen in the Introduction, up to a dilatation and a shift of the set of permutation vectors, this only amounts to relaxing the integer constraints on permutation vectors. The problem then becomes min

{1 T ⇡=0, k⇡k 2 =1} ⇡ T L A ⇡
This relaxed problem is an eigenvector problem. Finding the minimum over normalized vectors

x yields the eigenvector associated to the smallest eigenvalue of L A , but the smallest eigenvalue, 0, is associated with the eigenvector 1, from which we cannot recover any permutation.

However, if we restrict x to be orthogonal to 1, the solution is the second smallest eigenvector, called the Fiedler vector. A permutation is recovered from this eigenvector by sorting its coefficients: given x =( x 1 ,x 2 ,...,x n ), the algorithm outputs a permutation ⇡ such that

x ⇡(1)  x ⇡(2)  ...  x ⇡(n)
. This procedure is summarized in Algorithm 1.1, repeated here as Algorithm 2.1.

Algorithm 2.1 Spectral ordering [Atkins et al., 1998]

Input: Connected similarity matrix A 2 R n⇥n 1: Compute Laplacian L A = diag(A1) A 2: Compute second smallest eigenvector of L A , x ⇤ 3: Sort the values of x ⇤ Output: Permutation ⇡ : x ⇤ ⇡(1)  x ⇤ ⇡(2)  ...  x ⇤ ⇡(n)
In fact, [Atkins et al., 1998] showed that under the assumption that A has a serial structure, Algorithm 2.1 solves the seriation problem exactly, i.e., recovers the order ⇡ such that A ⇡(i)⇡(j) Once the reads are reordered, we can sequentially compute their exact positions (basepair coordinate of their left end on the genome) and orientation. We assign position 0 and strand "+" to the first read, and use the overlap information (position of the overlap on each read and mutual orientation) to compute the second read's position and orientation, etc. More specifically, when computing the position and orientation of read i, we use the information from reads i 1,...,i c to average the result, where c roughly equals the coverage, as this makes the layout more robust to misplaced reads. Note that overlappers relying on hashing, such as minimap and MHAP, do not generate alignments but still locate the overlaps on the reads, making this positioning step possible. Thanks to this "polishing" phase, we would still recover the layout if two neighboring reads were permuted due to consecutive entries of the sorted Fiedler vector being equal up to the eigenvector computation precision, for example.

Consensus generation

We built a simple assembler using this layout idea and tested its accuracy. It is partly inspired by the nanocorrect pipeline of Loman et al. [2015] in which reads are corrected using multiple alignments of all overlapping reads. These multiple alignments are performed with a Partial Order Aligner (POA) [Lee et al., 2002] multiple-sequence alignment engine. It computes a consensus sequence from the alignment of multiple sequences using a dynamic programming approach that is efficient when the sequences are similar (which is the case if we trim the sequences to align their overlapping parts). Specifically, we used SPOA, a Single Instruction Multiple Data implementation of POA developed in [START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF].

The key point is that we do not need to perform multiple alignment using all reads, since we already have a layout. Instead, we can generate a consensus sequence for, say, the first 3000 bp of the genome by aligning the parts of the reads that are included in this window with SPOA, and repeat this step for the reads included in the window comprising the next 3000 bp of the genome, etc. In practice, we take consecutive windows that overlap and then merge them to avoid errors at the edges, as shown in Figure 2.2. The top of the figure displays the layout of the reads broken down into three consecutive overlapping windows, with one consensus sequence generated per window with SPOA. The final assembly is obtained by iteratively merging the window k+1 to the consensus formed by the windows 1,...,k.

The computational complexity for aligning N sequences of length L with POA, with an average divergence between sequences ✏,isroughlyO(mN L 2 ), with m ' (1 + 2✏).W i t h10% of errors, m is close to 1. If each window of size L w contains about C sequences, the complexity of building the consensus in a window is O(mCL 2 w ). We compute L g /L w consensus windows, with L g the length of the genome (or contig), so the overall complexity of the consensus generation is O(mCL g L w ). We therefore chose in practice a window size relatively small, but large enough to prevent mis-assemblies due to noise in the layout, L w =3kbp.

Overlap-based similarity and repeats handling

In practice, we build the similarity matrix A as follows. Given an overlap found between the i-th and j-th reads, we set A ij equal to the overlap score (or number of matches, given in tenth column of minimap or fourth column of MHAP output file). Such matrices are sparse: a read overlaps with only a few others (the number of neighbors of a read in the overlap graph roughly equals the coverage). There is no sparsity requirement for the algorithm to work, however sparsity lowers RAM usage since we store the n ⇥ n similarity matrix with about n ⇥ C non-zero values, with C the coverage. In such cases, the ordered similarity matrix is band diagonal.

Unfortunately, the correctly ordered (sorted by position of the reads on the reference sequence) similarity matrix contains outliers outside the main diagonal band (see Figure 2 that corrupt the ordering. These outliers are typically caused by either repeated subsequences or sequencing noise (error in the reads and chimeric reads), although errors in the similarity can also be due to hashing approximations made in the overlap algorithm. We use a threshold on the similarity values and on the length of the overlaps to remove them. The error-induced overlaps are typically short and yield a low similarity score (e.g., number of shared min-mers), while repeat-induced overlaps can be as long as the length of the repeated region. By weighting the similarity, the value associated to repeat-induced overlaps can be lowered. Weighting can be done with, e.g., the -weighted option in MHAP to add a tf-idf style scaling to the MinHash sketch, making repetitive k-mers less likely to cause a match between two sequences, or with default parameters with minimap.

In the Supplementary Material presented in Chapter A, we describe experiments with real, Most outliers can be similarly removed for real and synthetic data from S. cerevisiae,a l t h o u g h a few outliers, probably harboring telomeric repeats, remain at the ends of chromosomes after thresholding.

There is thus a tradeoff to be reached depending on how many true overlaps one can afford to lose. With sufficient coverage, a stringent threshold on overlap score and length will remove both repeat-induced and error-induced overlaps, while still yielding a connected assembly graph. Otherwise, aggressive filtering will break the similarity graph into several connected components. In such a case, since the spectral algorithm only works with a connected similarity graph, we compute the layout and consensus separately in each connected component, resulting in several contigs. To set the threshold sufficiently high to remove outliers but small enough to keep the number of contigs minimal, we used a heuristic based on the following empirical observation, illustrated in Supplementary Figure A.4. The presence of outliers in the correctly (based on the positions of the reads) ordered band diagonal matrix imparts an increased bandwidth (maximum distance to the diagonal of non zero entries) on the matrix reordered with the spectral algorithm. We can therefore run the spectral algorithm, check the bandwidth in the reordered matrix, and increase the threshold if the bandwidth appears too large (typically larger than twice the coverage).

In practice, we chose to set the threshold on the overlap length to 3.5kbp, and removed the overlaps with the lowest score [in the first 40%-quantile (respectively 90% and 95%) for C60X (resp. 60XC100X and C 100X)]. As indicated in Algorithm 2.2, we let these threshold values increase if indicated by the bandwitdh heuristic.

Finally, we added a filtering step to remove reads that have non-zero similarity with several sets of reads located in distant parts of the genome, such as chimeric reads. These reads usually overlap with a first subset of reads at a given position in the genome, and with another distinct subset of reads at another location, with no overlap between these distinct subsets. We call such reads "connecting reads", and they can be detected from the similarity matrix by computing, for each read (index i), the set of its neighbors in the graph N i = {j : A ij > 0}. The subgraph represented by A restricted to N i is either connected (there exists a path between any pair of edges), or split into separate connected components. In the latter case, we keep the overlaps between read i and its neighbors that belong to only one of these connected components (the largest one). Reorder A with spectral algorithm (Algorithm 2.1)

6:

if bandwidth of A reordered 2⇥ Coverage then 7:

set higher threshold on A and try again 8:

end if

9:

Compute layout from the ordering found and overlaps 10:

Partition the length of the contig into small windows 11:

Compute consensus in each window with SPOA

12:

Merge consecutive windows with SPOA 13: end for Output: Contig consensus sequences

Results

Data

We tested this pipeline on ONT and PacBio data. The bacterium Acinetobacter baylyi ADP1

and the yeast Saccharomyces cerevisiae S288C were sequenced at Genoscope with Oxford Nanopore's MinION device using the R7.3 chemistry, together with an additional dataset of S. cerevisiae S288C using the R9 chemistry. Only the 2D high quality reads were used.

The S. cerevisiae S288C ONT sequences were deposited at the European Nucleotide Archive (http://www.ebi.ac.uk/ena) where they can be accessed under Run accessions ERR1539069 to ERR1539080. We also used the following publicly available data: ONT Escherichia coli by Loman et al.

Layout

Bacterial genomes minimap was used to compute overlaps between raw reads (we obtained similar results with MHAP and DALIGNER). The similarity matrix preprocessed as detailed in Section 2.2.3 yielded a few connected components for bacterial genomes. The reads were successfully ordered in each of these, as one can see in Figure 2.5 for E. coli,a n di nF i g u r eA.6 for the other datasets.

Figure 2.5: Ordering of the reads computed with the spectral algorithm vs true ordering (obtained by mapping the reads to the reference genome with GraphMap) for the E. coli ONT dataset. All contigs are artificially displayed on the same plot for compactness. There are two equivalent correct orderings for each contig : (1,2,...,n) and (n, n-1, ..., 1), both yielding the same 2-SUM score (2.1) and leading to the same consensus sequence (possibly reverse complemented).

Eukaryotic genome

For the S. cerevisiae genome, the threshold on similarity had to be set higher than for bacterial genomes because of a substantially higher number of repetitive regions and false overlaps, leading to a more fragmented assembly. Most of them are correctly reordered with the spectral algorithm, see Figure 2.6 and Supplementary Figure A.7.

Consensus

Recovering contiguity

Once the layout was established, the method described above was used to assemble the contigs and generate a consensus sequence. For the two bacterial genomes, the first round of layout produced a small number of connected components, each of them yielding a contig. Sufficient overlap was left between the contig sequences to find their layout with a second iteration of the algorithm and produce a single contig spanning the entire genome. The number of contigs in the yeast assemblies can be reduced similarly. The fact that the first-pass contigs overlap even though they result from breaking the similarity graph into several connected components might seem counter-intuitive at first sight. However, note that when cutting an edge A ij results in the creation of two contigs (one containing i and the other j), the sequence fragment at the origin of the overlap between the two reads is still there on both contigs to yield an overlap between them in the second iteration. Alternatively, we found the following method useful to link the contigs' ends: 1. extract the ends of the contig sequences, 2. compute their overlap with minimap, 3. propagate the overlaps to the contig sequences, 4. use miniasm with all pre-selection parameters and thresholds off, to just concatenate the contigs (see Supplementary Material §A.5).

Consensus quality evaluation

We first investigated the quality of the consensus sequences derived in each window. Figures 2.8 We also observed that repeats are often positioned at the edge between two contigs, though this does not seem to be the determinant factor. The bottom plots represent the error-rate in the windows against their estimated coverage, defined as the total length of sequences used to perform the multiple alignment in the window normalized by the length of the consensus sequence. Overall, one can see that the windows with high error rate are the ones with low coverage. Nevertheless, especially for the yeast genomes, there are also several windows with high values for both error-rate and coverage. Manual inspection of these reveals that they usually do not span repeated regions, but their high error-rates arise from imperfections in the layout. (c,d) panel represents the error-rates in consensus windows versus the coverage of the windows. The error-rate was computed with the errorrates.py script from samtools, using the mapping obtained from GraphMap.

We then compared our results to those obtained with other long reads assemblers : Miniasm, Canu and Racon [START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF]. Racon takes a draft assembly, the raw reads, and a mapping of the reads to the draft assembly as input. We used it with the draft assembly produced by Miniasm (as done by [START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF]). We label this method "Miniasm+Racon" in our results. We also used Racon with the draft assembly derived by our method ("Spec-tral+Racon" method), using Minimap to map the raw reads to the draft assemblies before using Racon. Racon's use here can be seen as a polishing phase for the sequences outputted by the spectral method and Miniasm. To keep both assemblers on an equal footing, we compared Spectral+Racon to two iterations of Miniasm+Racon (since one pass of Miniasm does not implement any consensus). A summary of assembly reports generated with DNAdiff [Kurtz et al., 2004]andQUAST [Gurevich et al., 2013] are given in Table 2.1 and Supplementary Table A.3.

Briefly, the assemblies displayed between 98% and 99% average identity to their reference genome, with errors mostly consisting in deletions. Misassemblies were rare in reconstructed bacterial genomes but more frequent in assembled yeast genomes, where they mostly consisted in translocations and relocations caused by either deletions and/or misplaced reads in the layout. Canu clearly outperforms the spectral method on PacBio data, while both assemblers yield comparable results on the ONT datasets.

Optical mapping

After the first iteration of the bacterial genome assembly pipeline, overlaps between the firstpass contigs were sufficient to find their layout. It should be anticipated however that not all overlaps might be apparent in some cases, e.g. if too many reads were removed during the preprocessing step. One attractive option is to use optical mapping [Aston et al., 1999]tolayout the contigs. We had such an optical map available for the A. baylyi genome, and implemented the algorithm of Nagarajan et al. [2008] to map the contigs to the restriction map, which led to the same layout as the one identified from our two-round assemblies (data not shown), thus providing a "consistency check" for the layout. We suggest in Table 2.2 and Figure 2.9 that optical maps could be particularly valuable for the ordering of contigs from more structurally complex eukaryotic genomes such as S. cerevisiae.

Table 2.2 displays assembly results for the following experiment. We divided a set of reads from S. cerevisiae into subsets, according to the chromosome membership of each read (obtained by mapping the reads to a reference genome). We then ran the method on each chromosomespecific dataset separately. The assembled contigs were evaluated with QUAST and DNAdiff for each chromosome (only a subset of the QUAST descriptive statistics is shown here). This experiments sheds light on how our method would behave if there were no repeats between chromosomes, or if we knew to which chromosomes some reads belong to thanks to, e.g., optical mapping. Figure 2.9 provides results from another experiment designed to evaluate the extent to which optical mapping could improve long-range anchoring of the 127 S. cerevisiae ONT R7.3 contigs and provide an alternative consistency check of the assembly. A restriction map was generated in silico from the reference S. cerevisiae genome with the BamHI restriction site (GGATCC), yielding one map per chromosome. Note that this simulated optical map represents a best-case scenario since real optical measurements lack some precision and are obtained through an error-prone assembly process. We used the same algorithm to layout the contigs with optical mapping as we had with the A. baylyi genome [Nagarajan et al., 2008]. Some contigs were correctly mapped by this process, while some others were not. Figure 2.9 shows histograms of the correctly and mis-mapped contigs according to the number of occurrences of the restriction site in the contigs, and to the length of the contigs. We observe that all contigs longer than 60kbp are correctly mapped.

Discussion

We have shown that seriation based layout algorithms can be successfully applied to de novo genome assembly problems, at least for genomes harboring a limited number of repeats.

In a similar vein to the recent report about the miniasm assembly engine [START_REF] Li | Minimap and miniasm: fast mapping and de novo assembly for noisy long sequences[END_REF], our work confirms that the layout of long reads can be found without prior error correction, using only overlap information generated from raw reads by tools such as minimap, MHAP or DALIGNER. However, unlike miniasm, which does not derive a consensus but instead concatenates the reads into a full sequence, we take advantage of read coverage to produce contigs with a consensus quality on par with that achieved by assembly pipelines executing dedicated error-correction steps. The results of Table 2.1 appear promising. For example, our assembler combined with Racon yields among the highest average identities with the reference for the ONT datasets. In terms of speed however, our pipeline is clearly outperformed by Miniasm, but also by Miniasm+Racon, the latter improving overall accuracy. Still, compared to approaches implementing error correction steps, we gain significant speed-ups by highly localizing the error correction and consensus generation processes, which is made possible by knowledge of the layout. We believe that tools such as Miniasm and Racon are implemented in a much more efficient way than our own, but the layout method itself is efficient (see Supplementary Table A.1) and is known to be scalable as it relies on the same algorithmic core as Google's

PageRank.

The main limitation of our layout algorithm is its sensitivity to outliers in the similarity matrix, hence the need to remove them in a pre-processing phase. Higher coverage and quality of the input reads, both expected in the near future, would likely improve the robustness of our pipeline. Still, for eukaryotic genomes, we found that some outliers require additional information to be resolved (see Supplementary Figure A.3), which could be provided in the future by extracting topological information from the assembly graph.

In the meantime, our pipeline behaves like a draft generating assembler for prokaryotic genomes, and a first-pass unitigger for eukaryotic genomes. Importantly, the overall approach is modular and can integrate other algorithms to increase layout robustness or consensus quality, as illustrated here by the integration of Racon as an optional polishing module.

Our original contribution here consists in the layout computation. The spectral OLC assembler we built on top of it could be enhanced in many ways. We have shown that the spectral algorithm is suited to find the layout for bacterial genomes, even though there is room left for performance improvements on repeat-rich eukaryotic genomes.

For these eukaryotic genomes, it could make sense to use the spectral algorithm jointly with other assembly engines (e.g. Miniasm or Canu), to check the consistency of connected components before they are assembled. Our consensus generation method is coarse-grained for now and does not take into account statistical properties of ONT sequencing errors. Nevertheless, the three components (O, L and C) of the method being independent, an external and more refined consensus generation process could readily be plugged after the overlap and layout computations to further improve results and increase accuracy.

Chapter 3

Multi-dimensional Spectral Ordering : Reconstructing Linear Orderings via Spectral Embedding

In the previous chapters, we have observed that due to repeats, the spectral method (Algorithm 1.1) fails to reorder full similarity matrices correctly into a single contig (see Figure 1.7b).

Yet, Algorithm 2.2, at the core of the method presented in Chapter 2, uses a simple iterative thresholding procedure leveraging the fact that the largest overlaps are scarcely due to repeats to adapt Algorithm 1.1. It yields correct but fragmented assemblies.

In this chapter, we explore an extension of the spectral method, that was at first motivated by the following experimental observation. While Figure 1.7b plots the first (non-trivial)

eigenvector of the Laplacian -the Fiedler vector, we can also take a look at the following eigenvectors. For instance, we can make a 3d scatter plot of the three eigenvectors associated to the three smallest non-zero eigenvalues. Interestingly, the points in this 3d scatter plot are roughly distributed along a curve with linear pieces bent in some points. Recall Theorem 3.2.1.

It states that if there is an ordering of the points such that the pairwise similarity decreases within their distance along this ordering, then the spectral method finds it. These assumptions mean that we can embed the points on a line such that the similarity is monotonic with the distance within the line. The Fiedler vector then provides such a linear embedding (it is a 1d embedding of the points, with one real value per coordinate i 2 [n]). Imagine we start with data satisfying the assumptions, but we add similarity between the first and the last elements in the chain so that the assumptions no longer hold. In the original linear embedding, the first and the last elements have high similarity but are placed far apart on the line. However, if we add one dimension to the embedding (an additional degree of freedom), and place ourselves on a plane, we can bend the line so that the first and the last elements are close to each other.

Specifically, we can obtain a circular embedding such that two elements that are nearby on the

Introduction

At the risk of being redundant, let us recall the seriation problem introduced in Chapter 1, before we present its generalization to circular orderings.

The seriation problem seeks to recover a latent ordering from similarity information. We typically observe a matrix measuring pairwise similarity between a set of n elements and assume they have a serial structure, i.e. they can be ordered along a chain where the similarity between elements decreases with their distance within this chain. In practice, we observe a random permutation of this similarity matrix, where the elements are not indexed according to that latent ordering. Seriation then seeks to find that global latent ordering using only (local) pairwise similarity.

Yet, in some applications, the latent ordering is circular. For instance, in de novo assembly of bacterial genomes, such as the E. coli and A. baylyi genomes encountered in Chapter 2,o n e has to reorder DNA fragments sub-sampled from a circular genome. The graphic illustration of de novo assembly shown in Chapter 1,F i g u r e1.4 was adequate for a linear strand of DNA, but Figure 3.1 is more appropriate for a circular genomes. The main structural hypothesis on similarity matrices related to seriation is the concept of R-matrix, which we have introduced in Chapter 1 and repeat here, together with its circular counterpart.

Definition 3.1.1. We say that A 2 S n is a R-matrix (or Robinson matrix) if it is symmetric and satisfies A i,j  A ik ,f o ra l lt r i p l e t so fi n d i c e s( i , j , k )s u c ht h a t|i j| |i k|.

Definition 3.1.2. We say that A 2 S n is a circular R-matrix if it is symmetric and satisfies In what follows, we write L n R (resp., C n R ) the set of R (resp., circular-R) matrices of size n, and P n the set of permutations of n elements. A permutation can be represented by a vector ⇡ (lower case) or a matrix Π 2 {0, 1} n⇥n (upper case) defined by Π ij =1iff ⇡(i)=j,and⇡ = Πg where g =( 1 ,...,n) T . We refer to both representations by P n and may omit the subscript n whenever the dimension is clear from the context. We say that A 2 S n is pre-L R (resp., pre-C R ) if there exists a permutation Π 2 P such that the matrix ΠAΠ T (whose entry (i, j) is In Section 3.2, we review the Spectral Ordering algorithm and the Laplacian Embedding used in Spectral Clustering. We mention graph-walk perspectives on this embedding and how this relates to dimensionality reduction techniques. Finally, we recall how these perspectives relate the discrete Laplacian to continuous Laplacian operators, providing insights about the curve structure of the Laplacian embedding through the spectrum of the limit operators. These asymptotic results were used to infer circular orderings in a tomography application in, e.g., Coifman et al. [2008]. In Section 3.3, we evidence the filamentary structure of the Laplacian Embedding, and provide theoretical guarantees about the Laplacian Embedding based method for Circular Seriation. We then propose a method in Section 3.4 to leverage the multidimensional Laplacian embedding in the context of Linear Seriation and Circular Seriation. In Section 3.5, we show that a perturbation analysis similar to that existing for Linear Seriation can be applied to Circular Seriation. We eventually present numerical experiments in Section 3.6 to illustrate how the spectral method gains in robustness by using a multidimensional Laplacian embedding.

A i,j  A ik ,
A ⇡(i),⇡(j) )i si nL R (resp., C R ). Given

Related Work

Let us recall the highlights of the spectral relaxation, which is the starting point of this work, before we review definitions and results involving higher-dimensional Laplacian embeddings.

Spectral Ordering for Linear Seriation

Linear Seriation can be addressed with a spectral relaxation of the 2-SUM combinatorial problem, minimize

P n i,j=1 A ij |⇡ i ⇡ j | 2 such that ⇡ 2 P n (2-SUM)
Intuitively, the optimal permutation compensates high A ij values with small |⇡ i ⇡ j | 2 ,thuslaying similar elements nearby. As we have seen in Section 1.2.2, for any

f = f (1),...,f(n) T 2
R n , the objective of 2-SUM can be written as a quadratic,

P n i,j=1 A ij |f (i) f (j)| 2 = f T L A f (3.1)
where

L A , diag(A1) A is the graph-Laplacian of A.F r o m( 3.1), L A is positive-semi-definite
for A having non-negative entries, and 1 =(1,...,1) T is an eigenvector associated to 0 =0.

The spectral method relaxes the 2-SUM problem by dropping the integer constraint on permutation vectors ⇡ 2 P n and enforcing only norm and orthogonality constraints, k⇡k =1 ,

⇡ T 1 =0,t oa v o i dt h et r i v i a ls o l u t i o n s⇡ =0and ⇡ / 1. It results in, minimize f T L A f such that kf k 2 =1,f T 1 =0. (Relax. 2-SUM)
This is an eigenvalue problem on L A solved by f (1) , the eigenvector associated to 1 0 the second smallest eigenvalue of L A . If the graph defined by A is connected (which we assume further) then 1 > 0.F r o mf (1) , one can recover a permutation by sorting its entries. The resulting algorithm, presented in Chapter 1 and applied to de novo assembly in Chapter 2,i s recalled here in Algorithm 3.1. We also recall a key theoretical result related to it. For pre-L R matrices, Linear Seriation is equivalent to 2-SUM [START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF], and can be solved with Algorithm 3.1 [Atkins et al., 1998], as stated in Theorem 3.2.1. Algorithm 3.1 Spectral ordering [Atkins et al., 1998]

Input: Connected similarity matrix A 2 R n⇥n 1: Compute Laplacian L A = diag(A1) A 2: Compute second smallest eigenvector of L A , f 1 3: Sort the values of f 1 Output: Permutation : f 1 ( (1))  ...  f 1 ( (n))
Theorem 3.2.1 (Atkins et al. [1998]). If A 2 S n is a pre-L R matrix, then Algorithm 3.1 recovers ap e r m u t a t i o nΠ 2 P n such that ΠAΠ T 2 L n R ,i . e . ,i ts o l v e sLinear Seriation.

Laplacian Embedding

Let 0= 0 < 1  ...  n 1 be the eigenvalues of L A ,a n dΦ =( 1,f 1 ,...,f n 1 ) the matrix whose column j is the eigenvector corresponding to the j-th smallest eigenvalue, which is the j 1 smallest, non-zero eigenvalue j 1 . We have the following decomposition, L A = ΦΛΦ T , with Λ , diag ( 0 ,..., n 1 ).

y i = f 1 (i),f 2 (i),...,f d (i) T 2 R d , for i =1,...,n. (d-LE)
The d-LE solves the following embedding problem, which is a generalization of 2-SUM to multi-dimensions, minimize

P n i,j=1 A ij ky i y j k 2 2 such that Φ = ⇣ y T 1 ,...,y T n ⌘ T 2 R n⇥d , ΦT Φ = I d , ΦT 1 n = 0 d (d-2SUM)
Indeed, thanks to the ortho-normality constraints, the objective of d-2SUM can also be written as a quadratic,

minimize Tr ⇣ ΦT L A Φ⌘ such that Φ 2 R n⇥d , ΦT Φ = I d , ΦT 1 n = 0 d (d-2SUM')
The reader can find a detailed derivation in Belkin and Niyogi [2003] could be used as alternatives to embed the data in a way that intuitively preserves the latent ordering. However, guided by the generalization of Algorithm 3.1 and theoretical results that follow, we restrict ourselves to the Laplacian embedding.

Normalization and Scaling

Several variants of the Laplacian (and resulting spectral embeddings d-LE) exist in the literature, leading to diverse interpretations and experimental behaviors.

Given the weighted adjacency matrix W 2 S n of a graph, its Laplacian reads L = D W , where D = diag(W 1) has diagonal entries d i = P n j=1 W ij (degree of i). Normalizing W ij by p d i d j or d i leads to the normalized Laplacians,

L sym = D 1/2 LD 1/2 = I D 1/2 WD 1/2 (symmetric) L rw = D 1 L = I D 1 W (random-walk)
They correspond to graph-cut normalizations (normalized cut or ratio cut). Moreover, L rw has a Markov chain interpretation, where a random walker on edge i jumps to edge j from time t to t +1 with transition probability P ij , W ij /d i . It has connections with diffusion processes, governed by the heat equation @Ht @t = ∆H t , where ∆ is the Laplacian operator, H t the heat kernel, and t is time [Qiu and Hancock, 2007]. These connections lead to diverse Laplacian embeddings backed by theoretical justifications, where the eigenvectors f rw k of L rw are sometimes scaled by decaying weights ↵ k (thus emphasizing the first eigenvectors),

ỹi = ↵ 1 f rw 1 (i),...,↵ d 1 f rw d (i) T 2 R d , for i =1,...,n. ((↵,d ) -LE)
Laplacian eigenmaps [Belkin and Niyogi, 2003] is a nonlinear dimensionality reduction technique based on the spectral embedding of L rw (((↵,d ) -LE) with ↵ k =1for all k). Specifically, given points x 1 ,...,x n 2 R d , the method computes a heat kernel similarity matrix

W ij =e x p kx i x j k 2 /
t and outputs the first eigenvectors of L rw as a lower dimensional embedding. The choice of the heat kernel is motivated by connections with the heat diffusion process on a manifold, a partial differential equation involving the Laplacian operator. This method has been successful in many machine learning applications such as semi-supervised classification [Belkin and Niyogi, 2004] and search-engine type ranking [START_REF] Zhou | Ranking on data manifolds[END_REF]]. Notably, it provides a global, nonlinear embedding of the points that preserves the local structure.

The commute time distance CTD(i, j) between two nodes i and j on the graph is the expected time for a random walker to travel from node i to node j and then return.

Link with Continuous Operators

In the context of dimensionality reduction, when the data points x 1 ,...,

x n 2 R D lie on a manifold M ⇢ R d of dimension K ⌧ D,
Γ = { (s) : s 2 [0, 1]},s a yx i = (i/n).
If their similarity measures their proximity along the curve, then the similarity matrix is a circular-R matrix if the curve is closed ( (0) = ( 1)), and a R matrix otherwise. Coifman et al. [2008] motivate a method for Circular Seriation with the spectrum of the Laplace-Beltrami operator ∆ Γ on Γ when Γ is a closed curve. Indeed, ∆ Γ is simply the second order derivative with respect to the arc-length s, ∆ Γ f (s)=f 00 (s) (for f twice continuously differentiable), and its eigenfunctions are given by,

f 00 (s)= f (s). (3.2)
With periodic boundary conditions, f (0) = f (1), f 0 (0) = f 0 (1), and smoothness assumptions, the first eigenfunction is constant with eigenvalue 0 =0 . The remaining eigenvalues m are double, associated to the eigenfunctions f m,cos and f m,sin given by, for m =1,...,1,

m =(2⇡m) 2 f cos m = cos (2⇡ms) f sin m =sin(2⇡ms)
Hence, the 2-LE given by the following equation should approximately lay the points on a circle, allowing for solving Circular Seriation [START_REF] Ronald R Coifman | Graph laplacian tomography from unknown random projections[END_REF],

f 1 (i),f 2 (i) ⇡ cos (2⇡s i ), sin (2⇡s i ) .
More generally, the 2d-LE given by the following equation is a closed curve in R 2d , f 1 (i),...,f 2d+1 (i) T ⇡ cos (2⇡s i ), sin (2⇡s i ),...,cos (2d⇡s i ), sin (2d⇡s i ) .

If Γ is not closed, we can also find its eigenfunctions. For instance, with Neumann boundary conditions (vanishing normal derivative),

f (0) = 1, f (1) = 0, f 0 (0) = f 0 (1) = 0,t h en o n -t r i v i a l
eigenfunctions of ∆ Γ , f m , associated to the eigenvalues m ,f o rm =1,...,1, are given by,

m =(⇡m) 2 f m = cos (⇡ms)
The 1-LE, f 1 (i) ⇡ cos (⇡s i ), respects the monotonicity of i, which is consistent with Theorem 3.2.1. Lafon [2004] invoked this asymptotic argument to solve an instance of Linear Seriation but seemed unaware of the existence of Atkin's Algorithm 3.1. Note that here too, the d-LE,

f 1 (i),...,f d (i) T ⇡ cos (⇡s i ),...,cos (d⇡s i )
follows a closed curve in R d , with endpoints.

These asymptotic results hint that the Laplacian embedding preserves the latent ordering of data points lying on a curve embedded in R D . However, these results are only asymptotic and there is no known guarantee for the Circular Seriation problem as there is for Linear Seriation.

Also, the curve (sometimes called filamentary structure) stemming from the Laplacian embedding has been observed in more general cases where no hypothesis on a latent representation of the data is made, and the input similarity matrix is taken as is (see, e.g., Diaconis et al. [2008] for a discussion about the horseshoe phenomenon).

Other embeddings

We focus on the Laplacian embedding since it naturally extends results from Atkins et al.

[1998]. However, other methods can produce low-dimensional embeddings from a similarity (or distance) matrix, such as Multi-Dimensional Scaling (MDS) [START_REF] Joseph | Multidimensional scaling[END_REF]. Classical -MDS uses the eigen-decomposition of the centered distance matrix, (it is also a spectral method). metric-MDS finds the embedding through the minimization of a stress function.

t-SNE [Maaten and Hinton, 2008] minimizes the divergence between similarity-based proba-

bilities to find a 2D or 3D embedding of the data. We experimentally compare the orderings found by our method when using these alternative embedding techniques.

Dimensionality reduction techniques, e.g., kernel PCA [START_REF] Schölkopf | Kernel principal component analysis[END_REF] and Locally Linear Embedding (LLE) [Roweis and Saul, 2000], take design matrices (high-dimensional embedding) as input, to produce the low-dimensional embedding, instead of distance/similarity matrices, hence we do not consider those in the following.

Ordering points lying on a curve

Existing approaches for Seriation rely on either 1D or 2D embeddings, sorting coordinates (1D), or angles between two coordinates (2D) to reorder the points. Friendly [2002]s o r t st h ea n g l e between the coordinates of the 2D-MDS embedding to perform Linear Seriation. Coifman et al. cell-cycle related problem, but with the 2D embedding given by MDS. We are not aware of any method using higher-dimensional embeddings under Linear Seriation or Circular Seriation assumptions.

Spectral properties of some (circular) Robinson matrices

We have claimed that the d-LE enhances the latent ordering of the data and we now present some theoretical evidences. We adopt a point of view similar to Atkins et al. [1998], where the feasibility of Linear Seriation relies on structural assumptions on the similarity matrix (L R ).

For a subclass C ⇤ R of C R (set of circular-R matrices), we show that the d-LE lays the points on a closed curve, and that for d =2 , the elements are embedded on a circle according to their latent circular ordering. This is a counterpart of Theorem 3.2.1 for Circular Seriation.

It extends the asymptotic results motivating the approach of Coifman et al. [2008], shifting the structural assumptions on the elements (data points lying on a curve embedded in R D )t o assumptions on the raw similarity matrix that can be verified in practice. Then, we develop a perturbation analysis to bound the deformation of the embedding when the input matrix is in C ⇤ R up to a perturbation. Finally, we discuss the spectral properties of some (non circular) L R -matrices that shed light on the filamentary structure of their d-LE for d>1.

For simplicity, we assume n , 2p +1 odd in the following. The results with n =2 p even are relegated to the Appendix Chapter B, together with technical proofs.

Circular Seriation with Symmetric, Circulant matrices

A = 0 B B B B B B B B B B @ b 0 b 1 b 2 ••• b 2 b 1 b 1 b 0 b 1 ••• b 3 b 2 b 2 b 1 b 0 ••• b 4 b 3 . . . . . . . . . . . . . . . . . . b 2 b 3 b 4 ••• b 0 b 1 b 1 b 2 b 3 ••• b 1 b 0 1 C C C C C C C C C C A . (3.3)
Where b is a vector of values of size p +1 (recall that n =2p +1). The circular-R assumption (Def 3.1.2) imposes that the sequence (b 0 ,...,b p+1 ) is non-increasing. We thus define the set

C ⇤ R of circulant matrices of C R as follows. Definition 3.3.1. AmatrixA 2 S n is in C ⇤ R iff it verifies A ij = b |i j| and b k = b n k for 1  k  n with (b k ) k=0,.
..,bn/2c a non-increasing sequence.

The spectrum of symmetric circulant matrices is known [Reichel and Trefethen, 1992, Gray et al., 2006[START_REF] Massey | Distribution of eigenvalues of real symmetric pavndromic toeplitz matrices and circulant matrices[END_REF], and for a matrix A of size n =2p +1, it is given by,

⌫ m = b 0 +2 P p k=1 b k cos 2⇡km/n y m,cos = 1 p n ⇣ 1, cos 2⇡m/n ,...,cos 2⇡m(n 1)/n ⌘ y m,sin = 1 p n ⇣ 1, sin 2⇡m/n ,...,sin 2⇡m(n 1)/n ⌘ . (3.4)
For m =1,...,p, ⌫ m is an eigenvalue of multiplicity 2 with associated eigenvectors y m,cos ,y m,sin .

For any m, (y m,cos ,y m,sin ) embeds the points on a circle, but for m>1, the circle is walked

(i) , tan 1 f 2 (i)/f 1 (i) + [f 1 (i) < 0]⇡ Output: Permutation : ✓( (1))  ...  ✓( (n))
Theorem 3.3.2. Given a permuted observation ΠAΠ T (Π 2 P)o fam a t r i xA 2 C ⇤ R ,t h e2 -LE maps the items on a circle, equally spaced by angle 2⇡/n,f o l l o w i n gt h ec i r c u l a ro r d e r i n gi n Π.H e n c e ,A l g o r i t h m3.2 recovers a permutation Π 2 P n such that ΠAΠ T 2 C ⇤ R ,i . e . ,i ts o l v e s Circular Seriation.

(Linear) Robinson Toeplitz matrices

In order to show Theorem 3.3.2, we have examined the spectrum of some circular-R matrices.

Although only the 2-LE appears in Theorem 3.3.2, it is interesting to see that for any d>1,the

d-LE of matrices in C ⇤
R is a curve. Let us investigate how the latent linear ordering of Toeplitz matrices in L R translates to the d-LE. Remark that from Theorem 3.2.1,t h e1 -LE suffices to solve Linear Seriation. Yet, for perturbed observations of A 2 L R ,t h ed -LE may be more robust to the perturbation than the 1-LE, as the experiments in Section 3.6 indicate. However, there is no closed form expression for the spectrum of (linear) R matrices in general, or even of Toeplitz R matrices, which are the analog of C ⇤ R for 2-SUM. Therefore, in the remainder of this Section, we will review spectral properties of specific standard Robinson matrices appearing in some applications, whose spectrum has been studied.

Tridiagonal

Spectral properties of the Laplacian

We have listed some spectral properties of typical similarity matrices. Let us conclude this section by remarking how the spectrum of a matrix relates to that of its Laplacian.

For circulant matrices A, L A and A have the same eigenvectors since L A = diag(A1) A = cI A, with c , P n 1 k=0 b k . For general symmetric Toeplitz matrices, this property no longer holds as c i = P n j=1 b |i j| varies with i. Yet, for fast decaying Toeplitz matrices, c i is almost constant except for i at the edges, namely i close to 1 or to n. Therefore, the eigenvectors of L A resemble those of A except for the "edgy" entries.

Note that using the eigenvectors of A to embed the points boils down to classical (or, nonmetric) multi-dimensional scaling (MDS) [START_REF] Joseph | Multidimensional scaling[END_REF]. MDS is a dimensionality reduction method aiming to find an embedding of points that preserves the pairwise distances.

Given a similarity matrix A, one can consider the distance matrix D , max(A) A,a n da p p l y MDS. Therefore, although the Laplacian embedding enjoys theoretical properties leading to Theorems 3.2.1, 3.3.2, in practice, classical MDS yields a similar embedding for R matrices with a fast decay.

Recovering Ordering on Filamentary Structure

We have seen that (some) similarity matrices A with a latent ordering lead to a filamentary d-LE. The d-LE integrates local proximity constraints together into a global consistent embedding. We expect isolated (or, uncorrelated) noise on A to be averaged out by the spectral picture. Therefore, we present Algorithm 3.3 that redefines the similarity S ij between two items from their proximity within the d-LE.

The Algorithm

Basically, our algorithm fits the points by a line locally, in the same spirit as LLE, which makes sense when the data lies on a linear manifold (curve) embedded in R K . Note that Spectral

Ordering (Algorithm 3.1) projects all points on a given line (it only looks at the first coordinates f 1 (i)) to reorder them. Our method does so in a local neighborhood, allowing for reordering points on a curve with several oscillations. We then run the basic Algorithms 3.1 (or 3.2 for Circular Seriation). Hence, the d-LE is eventually used to pre-process the similarity matrix.

Algorithm 3.3 Ordering Recovery on Filamentary Structure in R K . Input: As i m i l a r i t ym a t r i xA 2 S n , a neighborhood size k 2, a dimension of the Laplacian Embedding d.

1: Φ = ⇣ y T 1 ,...,y T n ⌘ T 2 R n⇥d d-LE(A)
. Compute Laplacian Embedding 2: Initialize S = I n . New similarity matrix 3: for i =1,...,n do 4:

V {j : j 2 k-NN(y i )} [ {i} . find k nearest neighbors of y i 2 R d 5:
w LinearFit(V )

. fit V by a line 6:

D uv |w T (y u y v )|,f o ru, v 2 V .
. Compute distances on the line 7:

S uv S uv + D 1 uv ,f o ru, v 2 V .
. Update similarity 8: end for 9: Compute ⇤ from the matrix S with Algorithm 3.1 (resp., Algorithm 3.2) for a linear (resp., circular) ordering. Output: A permutation ⇤ . In Algorithm 3.3, we compute a d-LE in line 1 and then a 1-LE (resp., a 2-LE) for linear ordering (resp., a circular ordering) in line 9. For reasonable number of neighbors k in the k-NN of line 4 (in practice, k = 15), the complexity of computing the d-LE dominates Algorithm 3.3.

We shall see in Section 3.6 that our method, while being almost as computationally cheap as the base Algorithms 3.1 and 3.2 (roughly only a factor 2), yields substantial improvements. In

Illustration of Algorithm 3.3

As a qualitative result, we provide a visual illustration of the method's behavior with a circular banded matrix in Figures 3. ran.

Perturbation analysis

The spectrum is a continuous function of the matrix. We can bound the deformation of the the concatenation of the two eigenvectors associated to the two smallest non-zero eigenvalues, written 1  2 for L.T h e n ,t h e r ee x i s t sa no r t h o n o r m a lr o t a t i o nm a t r i xO such that

kV 1 Ṽ1 Ok F p n  k Ak F min( 1 , 2 1 )
.

(3.7)

For circular matrices, we can derive a slightly finer result. ||p

⌧ (i) pi || 2  2 3/2 min( p 2||L H || 2 , ||L H || F ) min(| 1 |, | 2 1 |) , (3.8)
where 1 < 2 are the first non-zeros eigenvalues of L.

Proof. For a matrix V 2 R n⇥d , denote by

V 2,1 =s u p i=1,...,n V i 2 ,
where

V i are the columns of V . Because in R n we have || • || 1  || • || 2 , it follows that V 2,1  ||V i || i=1,...,n 2 = v u u t n X i=1 ||V i || 2 2  V F .
We apply [Yu et

|| Ṽ VO|| F  2 3/2 min( p 2||L H || 2 , ||L H || F ) min(| 1 |, | 2 1 |) . (3.9)
In particular we have These results bounding the perturbation of the embedding allow to find conservative guarantees of ordering recovery with Algorithm 3.2, based on simple geometric reasoning. Indeed, when all the points remain in a sufficiently small ball around their original position on the circle, Algorithm 3.2 can exactly find the ordering. Let us start with a geometrical lemma quantifying the radius of the ball around each (cos(✓ k ), sin(✓ k )) so that they do not intersect.

Ṽ VO 2,1  Ṽ VO F Ṽ VO 2,1  2 3/2 min( p 2||L H || 2 , ||L H || F ) min(| 1 |, | 2 1 |) Finally because A is a R-symmetric-
Lemma 3.5.3. For x 2 R 2 and ✓ k =2⇡k/n for k 2 N such that ||x (cos(✓ k ), sin(✓ k ))|| 2  sin(⇡/n) , (3.10)
we have

|✓ x ✓ k |  ⇡/n , where ✓ x = tan 1 (x 1 /x 2 ) + 1[x 1 < 0]⇡.
Proof. Let x that satisfies (3.10). Let's assume without loss of generality that ✓ k =0and ✓ x 0.A s s u m ea l s ot h a tx = e 1 +sin(⇡/n)u x where u is a unitary vector. A x for which ✓ x is maximum over these constrained is such that u x and x are orthonormal.

Parametrize u x = (cos( ), sin( )), because u x and x are orthonormal, we have cos( )= sin( ⇡/n). Finally since ✓ x 0, it follows that = ⇡/2+⇡/n and hence with elementary geometrical arguments ✓ x = ⇡/n. Proposition 3.5.4 (Exact circular recovery under noise in Algorithm 3.2). Consider a matrix à = Π T AΠ + H with A a R circular Toeplitz (Π is the matrix associated to the permutation )a n dH as y m m e t r i cm a t r i xs u c ht h a t

min( p 2||L H || 2 , ||L H || F )  2 3/2 sin(⇡/n)min(| 1 |, | 2 1 |) ,
where 1 < 2 are the first non-zeros eigenvalues of the graph Laplacian of Π T AΠ.D e n o t eb y ˆ the output of Algorithm 3.2 when having à as input. Then there exists a cyclic permutation ⌧ such that

ˆ = 1 ⌧ 1 . (3.11)
Proof. We have

Π T ÃΠ = A + Π T HΠ .
L is the graph Laplacian associated to A and L, the one associated to Ã.

(i) p ⌧ (i) || 2 < 2 3/2 min( p 2||L H ⇡ || 2 , ||L H ⇡ || F ) min(| 1 |, | 2 1 |)
, with H ⇡ = Π T HΠ, 1 < 2 the first non zero eigenvalues of A.

Graph Laplacian involve the diagonal matrix D H . In particular we have that

D H ⇡ = Π T D H Π. For the unnormalized Laplacian, it results in L H ⇡ = Π T L H Π. We hence have sup i=1,...,n ||p (i) p ⌧ (i) || 2 < 2 3/2 min( p 2||L H || 2 , ||L H || F ) min(| 1 |, | 2 1 |) sup i=1,...,n ||p i p ⌧ 1 (i) || 2 < sin(⇡/n) .
From Theorem 3.3.2, p i = cos(2⇡i/n) for all i. It follows that for any i

||p i cos(2⇡⌧ (i)/n)|| 2 < sin(⇡/n) .
Algorithm 3.2 recovers the ordering by sorting the values of

✓ i = tan 1 (p 1 i /p 2 i ) + 1[p 1 i < 0]⇡ ,
where pi =(p 1 i , p2 i ). Applying Lemma 3.5.3:

|✓ i 2⇡(⌧ 1 )(i)/n| < ⇡/n 8i 2 {1,...,n}, so that ✓ 1 ⌧ 1 (1)  •••  ✓ 1 ⌧ 1 (n) .
(3.12)

Finally ˆ = 1 ⌧ 1 .

Numerical Results

In this section, we present results from synthetic experiments where we seek to reorder pre-R matrices corrupted with noise. They quantify the performance gain achieved by using the d-LE instead of the 1-LE (or 2-LE, for Circular Seriation), in terms of correlation between the ground-truth permutation (for which the noiseless matrix is R), and the permutation found by the algorithms. Then, we set out to use our method to determine the layout of reads in de novo assembly, i.e., reordering overlap-based similarity matrices.

Synthetic Experiments

We performed synthetic experiments with noisy observations of Toeplitz matrices A, either linear (L R ) or circular (C ⇤ R ). We added a uniform noise on all the entries, with an amplitude parameter a varying between 0 and 5, with maximum value of the noise akAk F . The matrices A used are either banded (sparse), with linearly decreasing entries when moving away from the diagonal, or dense, with exponentially decreasing entries (KMS matrices). We used n = 500,

Genome assembly experiment : bacterial genomes with ONT long-reads

We tested the method on some of the datasets introduced in Chapter 2. Recall that in de novo genome assembly, a whole DNA strand is reconstructed from randomly sampled sub-fragments (called reads) whose positions within the genome are unknown. The genome is over-sampled so that all parts are covered by multiple reads with high probability. The Overlap-Layout-Consensus (OLC) assembly paradigm is based on three steps. First, compute the overlaps between all pairs of read. This provides a similarity matrix A, whose entry (i, j) measures how much reads i and j overlap (and is zero if they do not). Then, determine the layout from the overlap information, that is to say find an ordering and positioning of the reads that is consistent with the overlap constraints. This step, akin to solving a one dimensional jigsaw puzzle, is a key step in the assembly process. Finally, given the tiling of the reads obtained in the layout stage, the consensus step aims at determining the most likely DNA sequence that can be explained by this tiling. It essentially consists in performing multi-sequence alignments.

In the true ordering (corresponding to the sorted reads' positions along the genome), a

given read overlaps much with the next one, slightly less with the one after it, and so on, until a point where it has no overlap with the reads that are further away. This makes the read similarity matrix Robinson and roughly band-diagonal (with non-zero values confined to a diagonal band). Finding the layout of the reads therefore fits the Linear Seriation framework (or Circular Seriation for circular genomes, as illustrated in Figure 3.1). In practice however, there are some repeated sequences (called repeats) along the genome that induce false positives in the overlap detection tool [Pop, 2004], resulting in non-zero similarity values outside (and possibly far away) from the diagonal band. The similarity matrix ordered with the ground truth is then the sum of a Robinson band matrix and a sparse "noise" matrix, as in Figure 3.7a.

Because of this sparse "noise", the basic spectral Algorithm 3.1 fails to find the layout, as the quadratic loss appearing in 2-SUM is sensitive to outliers.

In Chapter 2, we have proposed the so-called bandwidth heuristic, which relies only on the baseline Algorithm 3.1 and iteratively breaks the overlap graph in connected components until the sub-components seem to contain no outlier. Instead, we show here that the simple multi-dimensional extension proposed in Algorithm 3.3 suffices to capture the ordering of the reads despite the repeats.

We used our method to perform the layout of an E. coli and A. baylyi bacterial genomes sequenced with the Oxford Nanopore Technology MinION device. Details on the data are given in Section 2.3.1 from Chapter 2. We computed the overlaps with the minimap2 dedicated software [Li, 2018], as detailed in Appendix B.2.1.

The method only worked with a sufficient threshold on the input similarity matrix in a pre-processing step. Here, we used 50% for E. coli dataset, and 70% for A. However, the sub-orderings could be unambiguously merged into one in a simple way described in Algorithm B.1, resulting in the orderings shown in Figures 3.7b and 3.7d. In practice, the threshold on the input similarity can be set as high as possible as long as the resulting suborderings can be merged into one single component (yielding a single contig). This criterion leads to the results presented here, where the bacterial genomes are correctly reordered.

The Kendall-Tau score between the ordering found and the one obtained by sorting the position of the reads along the genome (obtained by mapping the reads to a reference with minimap2) is of 99.5% for the E. coli dataset, and 99.3% for A. baylyi,u s i n gA l g o r i t h mB.2 to account for the circularity of the genome.

Genome assembly using Hi-C data

Although the output of a DNA sequencing experiment consists in a linear representation of the genome (a string, e.g., 'AAAT...GC'), or a collection of linear sequences (when there are several contigs or chromosomes), physical DNA has a non-linear spatial organization in the 3-D space (it can be thought of as a ball of yarn, or a plate of spaghetti). Hi-C is a chromosome conformation capture (3C) technique measuring the frequency of physical interactions between genomic loci that are nearby in 3-D space, though they can be separated by many nucleotides in the linear genome. Experiments indicate that the spatial proximity between genomic loci is not random, and provide valuable information in, e.g., gene identification and regulation We performed experiments using synthetic and real Hi-C data. The synthetic data include similarity matrices of four synthetic genomes of lengths 100, 150, 300 and 1000 bins. In the true ordering, these similarity matrices follow a power-law decay, 1/|i j +1| 2 , when moving away from the diagonal, with additive uniform noise.

There are also two synthetic similarity matrices modeling Hi-C data from a genome with multiple chromosomes, where the entries within a given chromosome (block of the matrix) are generated with the same rules as the single-stranded synthetic matrices described above, and the entries between distinct chromosomes are set with a low amplitude, sparse noise. The first one, called DL1, has 7 chromosomes of respective lengths 30, 40, 50, 70, 50, 40, 30 bins (total length 310 bins). The second one, called DL2, has 7 chromosomes of lengths 10, 20, 40, 80, 160, 320, 370 bins (total length 1000 bins).

Finally, we used four similarity matrices constructed from real data. The first one was built by mapping Hi-C reads of Plasmodium knowlesi to a reference genome split in 10kbp bins.

The reference genome is made of 14 chromosomes, and the resulting matrix is of size 2014.

The other matrices were built by mapping Hi-C reads of Spodoptera frugiperda to a genome assembly obtained with Pacbio long reads. The reference assembly is fragmented, and each of the three matrices Sf200, Sf669 and Sf846 is the restriction of the Hi-C reads that map to a given contig. However, these contigs may contain multiple chromosomes, or separated regions that

Single-stranded synthetic data

In Table 3.1, we provide the Kendall-Tau (written ⌧ ) correlation score between the true permutation and the ordering obtained with the spectral baseline method (Algorithm 1.1)andthe method introduced in this chapter (Algorithm 3.3), on the synthetic single-stranded frequency matrices of sizes 100, 150, 300, 1000. We also indicate the running time (ran on a 2014 Macbook Pro). We can see that although our method provides a marginal improvement on these matrices, the noise is sufficiently low for Algorithm 1.1 to be efficient.

Assembly of genomes with multiple chromosomes with Hi-C data

Let us consider the synthetic, multiple-chromosomes Hi-C data and the real data (which also contains separated fragments). For genomes with several chromosomes, we wish to find an ordering of the bins within any chromosome, rather than a global ordering with all chromosomes mixed. Yet, the chromosome assignment is not given in the Hi-C data. We can then attempt to cluster the bins into distinct chromosomes before reordering the elements in a given chromosome. If there was zero similarity between any two bins that span distinct chromosomes, then the input similarity matrix would be disconnected, and it would be trivial to find this cluster assignment. However, in practice we observe some high values of frequency of interactions between bins spanning distinct chromosomes, making the clustering step non trivial. In some cases, for instance here with the synthetic data and the Plasmodium knowlesi data, the user may know in advance the number of target chromosomes. In some others, for instance here with the Spodoptera frugiperda data, or in a de novo perspective, that number is unknown.

Methods for clustering and ordering

Let us comment on the spectral method for seriation and clustering.

f (i)=f (j)=1,thusf (i) f (j)=0,ori, j / 2 C k and f (i)=f (j)=0,thusthusf (i) f (j)=0, or i 2 C k ,j / 2 C k and A ij =0.
Therefore, all the products appearing in (3.1) are equal to zero. This is at the core of spectral clustering [START_REF] Von | A tutorial on spectral clustering[END_REF]. Therefore, for a matrix with K disconnected clusters, the K-LE only contains information regarding the clustering, but not the intra-cluster ordering, which is relegated to the higher-order eigenvectors. On the other hand, as we have seen in this chapter, if the similarity matrix is connected, then the eigenvalue 0 has a unique corresponding eigenvector, 1, and the Fiedler vector (the first non-zero eigenvector) is associated to the second smallest eigenvalue 1 > 0 and contains information about the ordering (remark that we have used the notation 1 for the second smallest eigenvector in this chapter, although the notation 2 > 0 is often used in the literature). When the similarity is "weakly" connected, 1 gets close to 0, the computation of the K-LE becomes less stable, and the ordering information contained in the K-LE gets diluted at the profit of clustering information. Therefore, we also include the t-SNE-embedding version of our method in the experiments, as t-SNE handles disconnected matrices seamlessly, where the clustering appears in the 2D embedding without losing the local serial structure, if any.

We consider the following methods to obtain a chromosome assignment and a reordering of the bins inside each chromosome. We denote the method presented in this chapter by mdso, standing for multi-dimensional spectral ordering.

Pre-processing. We found empirically that the following pre-processing of the similarity matrices enhanced the cluster structure and improved the results of the methods described below,

A ij P h2k-NN(i),l2k-NN(j) A hl |k-NN(i)||k-NN(j)| , (3.13)
where k-NN(i) are the k-nearest neighbors of i (the bins with the top k similarity values with i).

In practice we use k = 15. Appendix Figure B.8 illustrates the effect of this pre-preprocessing.

Spectral Clustering + Spectral Ordering (SC+SO). We first cluster the data from the similarity matrix (using no assumption regarding the Robinsonian structure) with spectral clustering, and then use Algorithm 1.1 in each cluster.

Spectral Clustering + mdso (SC+mdso). We first cluster the data from the similarity matrix (using no assumption regarding the Robinsonian structure) with spectral clustering, and then use Algorithm 3.3 in each cluster.

mdso. We only run Algorithm 3.3 on the input similarity, hoping that the d-LE will capture both clustering and intra-cluster ordering information, leading to a new similarity matrix S computed in Algorithm 3.3 that has connected components corresponding to the chromosomes.

tSNE-mdso. Same as the previous method, except that we use t-SNE to compute the embedding instead of the Laplacian embedding, with d =2 . The behaviour of t-SNE does not change whether the input similarity is connected or disconnected. If it is disconnected, it will simply find an embedding with separate clusters, but keep the intra-cluster structure.

Remark that the two first methods (SC+SO and SC+mdso) require the user to provide the number of desired clusters as input. Therefore, we only use them for the synthetic data and the Plasmodium knowlesi, for which we know the number of chromosomes.

Evaluation of clustering and sub-orderings

We evaluate the quality of the clustering with respect to the ground truth chromosome assignments with two scores. Given two partitions into K clusters, (Ω 1 ,...,Ω K ) and (C 1 ,...,C K ),

such that [ K k=1 Ω k = [ K k=1 C k = {1,...,N},a n d\ K k=1 Ω k = \ K k=1 C k = ;, the purity index defined by, purity(Ω,C)= 1 N K X k=1 max j=1,...,K |Ω k \ C j | (3.14)
where |•| is the cardinal of a set. It takes value between 0 and 1, and is equal to 1 if and only if both partitions are equal (the higher, the better). This metric can only be used when we compare two partitions with the same number of clusters. When using spectral clustering, the user specifies the number of clusters, hence we can always find a clustering with as many clusters as the ground truth. Still, when using mdso or tSNE-mdso, we do not control the number of clusters found by the method. Therefore, we also use the following metric measuring the distance between two partitions (Ω 1 ,...,Ω K ) and C 1 ,...,C 0 K (where K and K 0 may differ), as in Bach and Harchaoui [2008],

d 2 (Ω,C)= K + K 0 2 K X k=1 K 0 X k 0 =1 |Ω k \ C k 0 | 2 |Ω k ||C k 0 | (3.15)
It takes value between 0 and K+K 0 2 1 (and between 0 and K 1 if the two partitions have the same number of clusters) (the lower, the better).

Finally, given two raking (in the form of a permutation) ⇡ 1 and ⇡ 2 ,wesaythatapairi<jis concordant if the two rankings agree, namely, ⇡ 1 (i) > ⇡ 1 (j) and ⇡ 2 (i) > ⇡ 2 (j),or⇡ 1 (i) < ⇡ 1 (j) and ⇡ 2 (i) < ⇡ 2 (j). We say that they are discordant otherwise, namely, ⇡ 1 (i) > ⇡ 1 (j) and If we have clustered data, it makes no sense to compare global orderings including the concordance of pairs (i, j) where i and j are in two separate clusters. We therefore use the following definition of the weighted Kendall-Tau metric between two sets of local orderings in K clusters,

⌧ = P k ⌧ k n k (n k 1)/2 P k n k (n k 1)/2 (3.17)
where ⌧ k is the Kendall-Tau score between the two local orderings in the k-th cluster, and n k is the number of points within the k-th cluster.

Results on data with reference clustering

In Table 3.2 we compare these methods on the two synthetic datasets DL1 and DL2. We observe that on this data, the inter-chromosomes frequency is sufficiently low for all methods to recover the correct clustering. Notably, the spectral and t-SNE based embeddings used in the two versions of mdso contain both clustering and ordering information (multiple, separated filamentary structures). We illustrate this in Figure 3.8. The two first dimensions of the spectral embedding (Figure 3.8b) contain mostly clustering information, although zooming in allow to see that the points roughly follow a filament in each cluster. The higher order eigenvectors of the Laplacian (Figure 3.8c) contain partial ordering information. Note that Algorithm 3.3 leverages all these eigenvectors simultaneously in the line fitting procedure. We also display the sub-orderings found in the chromosomes for the mdso method in Figure 3.9. In Table 3.3, we provide results on the Plasmodium knowlesi data. Interestingly, for the tSNE-mdso method, the weighted Kendall-Tau (⌧ ) metric varies significantly depending on the random initialization of t-SNE, with values ranging from about 60% to 92%. After investigation, it appears that the low values of ⌧ are due to the fact that when the method finds a cluster which contains several non contiguous sets of bins (say, two distant chromosomes), even if the ordering within the two chromsomes is approximately correct, the way the sub-orderings are arranged together can make ⌧ vary. We illustrate these variations in Figure 3. ⌧ score is evidently due to the blue component spanning several chromosomes. In Table 3.3, we provide the results for tSNE-mdso in the following form : mean ± standard deviation, computed through 30 experiments with different random initialization of tSNE.

Results on real data with no reference clustering

Here, we present results on the Hi-C data from Spodoptera frugiperda where the reference genome used was a fragmented assembly from Pacbio reads, with some mis-assemblies. The similarity matrices used here may contain fragments to separate (like chromosomes), but we ignore their locations and their number. Hence, we cannot provide the number of cluster to spectral clustering, and we cannot assess the quality of the clustering found by mdso and tSNEmdso. Therefore, in Table 3.4, we only give the number of clusters and the weighted kendall-tau scores for mdso and t-SNE mdso, and we test the Spectral method without prior clustering.

The results for tSNE-mdso are also averaged over 30 experiments, as with the Plasmodium knowlesi data. In Appendix Figure B.10, we show the pre-processed similarity matrices (with a logarithmic colormap for ease of reading), and the resulting orderings obtained with mdso.

Finding circular orderings with single-cell Hi-C data

Single-cell Hi-C data allows to capture part of the 3D architecture of DNA in the nucleus of individual cells. The similarity matrices used here are not Hi-C frequency matrices as in profiles were derived for each cell (such as the frequency matrices appearing above), and a pairwise similarity between cells was computed, based on the similarity between their frequency interaction profiles. used in these methods.

Conclusion

In this chapter, we bring together results that shed light on the filamentary structure of the Laplacian embedding of serial data. It allows for tackling Linear Seriation and Circular Seriation in a unifying framework. Notably, we provide theoretical guarantees for Circular Seriation analog to those existing for Linear Seriation. These do not make assumptions about the underlying generation of the data matrix, and can be verified ap o s t e r i o r iby the practitioner. Then, we propose a simple method to leverage the filamentary structure of the embedding. It can be seen as a pre-processing of the similarity matrix. Although the complexity is comparable to the baseline methods, experiments on synthetic and real data indicate that this pre-processing substantially improves robustness to noise.

From a genome assembly application standpoint, generalizing the spectral Algorithm 3.1 to

Circular Seriation provides a sounder model for laying out circular bacterial genomes. However, in practice, the repeat-induced overlaps also thwart Algorithm 3.2. Still, letting additional degrees of freedom in the d-LE enables the serial structure of the data to stand out although the repeats constrain some elements to remain close to each other in the embedding. The algorithm we propose leads to single-contig layouts for bacterial genomes.

Introduction

In the seriation problem, we are given a similarity matrix between a set of n elements, which we assume to have a serial structure, i.e., which can be ordered along a chain where the similarity between elements decreases with their distance within this chain. Among the applications of seriation, ranging from fields such as archeology [Robinson, 1951] Here, S n denotes the set of real symmetric matrices of dimension n. In what follows, we write R ⇤ n the set of strong-R-matrices of size n,a n dP n the set of permutations of n elements. A permutation can be represented by a vector ⇡ (lower case) or a matrix Π 2 {0, 1} n⇥n (upper case) defined by Π ij =1iff ⇡(i)=j,a n d⇡ = Πg where g =(1,...,n) T . We refer to both representations by P n and may omit the subscript n whenever the dimension is clear from the context. We say that A 2 S n is pre-R ⇤ if there exists a permutation Π 2 P such that the matrix ΠAΠ T (whose entry (i, j) is A ⇡(i),⇡(j) )i sas t r o n g -R -matrix, and the seriation problem seeks to recover this permutation Π, i.e.,s o l v e find Π 2 P such that ΠAΠ T 2 R ⇤ (Seriation) in the variable Π 2 P. This is illustrated in Figure 4

A i,j B ⇡(i),⇡(j) (QAP(A,B))
with B ij = |i j| 2 . Laurent and Seminaroti [2015] showed that for pre-R ⇤ matrices, Seriation is equivalent to QAP(A,B) when B 2 R ⇤ n , i.e. when B has increasing values when moving away from the diagonal, and has constant values across a given diagonal (i.e. B is a Toeplitz matrix). This includes p-SUM problems, for p>0, corresponding to B ij = |i j| p . The case p =1is also known as the minimum linear arrangement problem (MLA) [George and Pothen, 1997]. For pre-R ⇤ matrices, these problems are all equivalent and can be solved by the spectral algorithm of Atkins et al. [1998], described in Algorithm 1.1. However, when A is not pre-R ⇤ ,t h eSeriation problem has multiple local solutions, and the spectral algorithm does not necessarily find a global optimum for 2-SUM,p-SUMorQAP(A,B) with B a Toeplitz, negated R matrix. In fact, these problems are NP-hard in general [Sahni and Gonzalez, 1976 In Section 4.2, we introduce the robust seriation problem, motivated by applications to genome assembly. We show that for DNA data obeying a simple model which takes repeats into account, robust seriation is equivalent to Robust 2-SUM, which is a QAP problem similar to 2-SUM, where the squared distance to the diagonal that appears in the loss function is truncated. This truncated quadratic can be relaxed as a Huber loss. We present experiments to compare existing and new algorithmic approaches to solve this problem on two datasets: synthetic data following our simple model, and real data from an E. coli genome sequenced with third generation sequencing tools.

Robust Seriation

Classical Seriation is written as a feasibility problem: find the permutation that reorders the input matrix into an Robinson matrix. When A is pre-R ⇤ ,s o l v i n g2-SUM yields this permutation. However, when A is not pre-R ⇤ ,t h em a t r i xA reordered using the permutation that minimizes 2-SUM may be far from being R. Robust seriation seeks to find the closest pre-R ⇤ matrix to A and reorder it, solving instead minimize kS ΠAΠ T k such that Π 2 P,S 2 R ⇤ .

(Robust Seriation)

where the variable Π 2 P is a permutation matrix, the variable S 2 R ⇤ is a strong-R-matrix, and the norm is typically either the l 1 norm on components or the Froebenius norm.

Application of Seriation to Genome Assembly

De novo genome assembly has been presented in the Introduction, Section 1.3.1, and repeatedly throughout the chapters of this manuscript. As a reminder, it aims to reconstruct a DNA strand from fragments (reads) randomly sampled throughout the genome (and whose position on the genome is unknown). A common method is to compute the overlaps between all pairs of read, providing a similarity matrix A, whose entry (i, j) measures how much reads i and j overlap (and is zero if they do not). Then, we can determine the layout from the overlap information, that is to say find an ordering and positioning of the reads that is consistent with the overlap constraints.

In the true ordering (corresponding to the sorted reads' positions along the genome), a

given read overlaps much with the next one, slightly less with the one after it, and so on, until a point where it has no overlap with the reads that are further away. This makes the read similarity matrix Robinson and roughly band-diagonal (with non-zero values confined to a diagonal band). Finding the layout of the reads therefore fits the Seriation framework. In practice, however, there are repeated sequences (repeats) along the genome that induce false positives in the overlap detection tool [Pop, 2004], resulting in non-zero similarity values outside (and possibly far away) from the diagonal band. The similarity matrix ordered with the ground truth is then the sum of a Robinson band matrix and a sparse "noise" matrix, as displayed in Recently, long-reads sequencers such as PacBio's SMRT and Oxford Nanopore Technology (ONT) spurred a renaissance in assembly by enabling sequencing reads over 10kbp (kilo basepairs) long, resolving many small repeats [Koren and Phillippy, 2015]. However, their error rate is high (⇠ 10%). Thus, many assemblers include a correction module in a preprocessing step, which can help in separating repeats when the repeated copies slightly differ [Pop, 2004]. They also use statistical models on the data generation in order to filter out the overlaps that are likely to be repeat-induced, and retrospectively inspect the overlap graph for potential errors in a greedy fashion, until the graph is "cleaned" and contains as few ambiguities for reconstruction as the model allows for [Koren et al., 2017[START_REF] Li | Minimap and miniasm: fast mapping and de novo assembly for noisy long sequences[END_REF]. When there are ambiguities, the ambiguous reads are removed and the resulting assembly is fragmented.

In contrast, the approach presented in Chapter 2 is simpler and more principled. Yet, the presence of repeats often corrupts the ordering, as we illustrate in Figure 4.2, and previously in Figure 1.7b. To overcome this issue, the threshold-based method also ends up removing overlaps from the graph. Although it does not explicitly winnow out the repeats with a dedicated module, it eventually yields fragmented assemblies.

Here, we seek to apply Robust Seriation to genome assembly, dealing with the repeats in a principled manner. To this end, let us introduce stylized matrices modeling overlap-based similarity matrices arising in genome assembly. We write M n ( ,s) the set of matrices in {0, 1} n⇥n that are the sum of a band matrix of bandwidth and a sparse out-of-band matrix with s non-zero elements, Definition 4.2.1. A 2 {0, 1} n⇥n belongs to M n ( ,s) iff it is symmetric and satisfies A ij =1for all (i, j) such that |i j|  ,a n dn n z (A)= n +(2n 1) 2 + s.

Here nnz(A) is the number of non-zero elements of A, and the first term in the sum is the total number of elements in the bands. This means in particular s  n 2 n +(2n 1)

2

(the total number of non-zeros cannot exceed n 2 ). In this setting, we wish to find an ordering in which most pairs of similar elements are nearby. The 2-SUM objective can perform poorly here, since it strongly penalizes orderings with non-zero values far away from the diagonal, even when there is a small number of them, as we can see in Figure 4.2. Reducing this penalty on outliers is the goal of the robust seriation methods detailed below.

Robust 2-SUM

Given A 2 S n , Robust Seriation seeks to find a pre-R ⇤ matrix that is as close to A as possible.

Instead of searching directly for a perturbation of A that is pre-R ⇤ , we search for a perturbation of A that yields a low 2-SUM score, solving minimize

P n i,j=1 S ij |⇡ i ⇡ j | 2 + kA Sk 1 such that ⇡ 2 P,S 2 S + . (R2S( ))
where S + is the set of symmetric matrices with non-negative entries, and we use the l 1 norm on the difference between A and S to enforce sparsity in errors. Here, is a parameter that controls the deviation of S from A. The sum is separable and the minimization in S is closed form. Indeed, for a given (i, j), the function

S ij ! S ij ∆ 2 ij + |S ij A ij | is piecewise linear, with slope ∆ 2 ij for S ij <A ij ,a n d∆ 2 ij + for S ij >A ij , and is therefore minimal at S ij = A ij if ∆ 2 ij  and S ij =0otherwise (recall that S ij is constrained to be non-negative). Hence, R2S( ) is equivalent to minimize P n i,j=1 A ij min( , |⇡ i ⇡ j | 2 ) such that ⇡ 2 P. (R2SUM( ))
in the variable ⇡ 2 P. We now show that for stylized genome assembly similarity matrices, if the number of reads spanning repeated regions is controlled, then solving R2SUM( ) also solves Robust Seriation.

Proposition 4.2.2. For s  s lim , (n 1) and A 2 S n ,i fA can be permuted to belong to M n ( ,s),i . e . ,i ft h e r ei sΠ 2 P n : ΠAΠ T 2 M n ( ,s),t h e nΠ solves both Robust Seriation and R2SUM( ) with parameter = 2 ,a n dt h e`1 norm in Robust Seriation.

Proof. Let ,s be two positive integers such that  n, s  (n 1).W i t h o u tl o s s of generality, assume that A 2 M( ,s), i.e., Π = I, the identity permutation (otherwise, we simply factor out the true permutation). First, let us observe that for = 2 , I is optimal for R2SUM( ). Indeed, since A 2 {0, 1} n⇥n , the objective in R2SUM( ) is the sum of min( 2 , |⇡ i ⇡ j | 2 ) over all indexes (i, j) such that A ij =1. This sum can be split into two terms,

f in = X (i,j):A ij =1 , |⇡ i ⇡ j | |⇡ i ⇡ j | 2 , f out = X (i,j):A ij =1 , |⇡ i ⇡ j |> 2 .
For Π = I, the number of terms in f in is maximized since A ij =1for all (i, j) such that |i j|  (A 2 M( ,s)). The sum of the number of terms in f in and f out is equal to nnz(A) and is invariant by permutation (therefore, the number of terms in f out is also minimized for Π = I) Since any term in f in is smaller than any term in f out , Π = I is optimal for R2SUM( ) with = 2 . Now, let us see that Π = I is also optimal for Robust Seriation. Given Π, optimizing over S in Robust Seriation yields S Π = Proj R ⇤ (ΠAΠ T ), the projection of ΠAΠ T onto the set of strong-R-matrices. Let us assume that we use the `1 norm in (Robust Seriation). Then, S Π , the projection in `1 norm of the binary matrix ΠAΠ T , is also binary, as we prove further in Lemma 4.2.3. A sparse, {0, 1} strong-R-matrix is necessarily of the form

8 > > < > > : S ij =1 if |i j|  k, S ij =0 if |i j| >k+1, S ij 2 {0,
(k)=0 .F o rk = , n in (k)=0and
n out (k)  s (it is equal to s minus the number of elements in the +1-th diagonal). Thus, Π = I is optimal, and k = .

Note that in practice, one has to chose the parameter without observing before trying to solve R2SUM( ). Yet, for matrices A satisfying the hypothesis of Proposition 4.2.2,t h e number of non-zero values of A (which is observed even when A is permuted) provides a way to estimate . We compute it as the smallest integer such that the number of non-zero elements in a band matrix of size is larger than nnz(A). Also remark that the proof of Proposition 4.2.2 is conservative: it only involves reasoning about the location of non-zero values of a vectorized version of ΠAΠ T . Permuting rows and columns of a matrix adds constraints on the locations of these non-zero values that we did not take into account.

Lemma 4.2.3. Given a binary symmetric matrix S 2 {0, 1} n⇥n ,i th a sab i n a r yp r o j e c t i o ni n`1 norm onto the set of strong-R-matrices, that is to say, there exists a solution R 2 {0, 1} n⇥n to the following problem, minimize

P n i,j=1 |R ij S ij | such that R 2 L R . (R-proj)
Proof. Consider a given diagonal 0  k  n 1 in the lower triangle. The strong-R constraints are lower and upper bounds on the values of R ij on the k-th diagonal. Let m k , min i,j : |i j|=k R ij ,andM k , max i,j : |i j|=k R ij . Recall that S has only ones and zeros on the k-th diagonal. From R-proj, R ij has values in [0,1]. Clearly, a solution of R-proj satisfies,

R ij = 8 > < > : M |i j| , if S ij =1 m |i j| , if S ij =0.
Let 0  p k  n k denote the number of ones on the k-th diagonal of S,and0  z k = n k p k the number of zeros on the k-th diagonal of S. Summing over all the diagonals of the matrix, the objective in R-proj can be written as,

kS Rk 1 = p 0 (1 M 0 )+z 0 (m 0 0) + 2 P n 1 k=1 p k (1 M k )+z k (m k 0) (4.1)
where we have separated the main diagonal from the others that are coupled with their symmetric. Now, we have that

0  m k  M k  1 for all 0  k  n 1. The strong-R constraints also require that M k  m k 1 for 1  k  n 1.
The minimizer of R-proj saturates these constraints (M k = m k 1 ), and equation (4.1) can finally be written as,

kS Rk 1 = p 0 (1 M 0 )+z 0 m 0 +2 P n 1 k=1 p k (1 m k 1 )+z k m k = p 0 (1 M 0 )+(z 0 2p 1 ) m 0 +2 P n 1 k=1 (z k p k+1 ) m k + P n 1 k=1 p k .
where by convention p n , 0. R-proj seeks to minimize this objective on the variables (M 0 ,m 0 ,m 1 ,...,m n 1 ),

under the constraints 1 M 0 m 0 , m k 1 m k for 1  k  n 1,a n dm k 0 for 0  k  n 1.
All in all, we can rewrite R-proj as a linear program over the variable x =(M 0 ,m 0 ,m 1 ,...,m n 1 ) 2 R n+1 , minimize c T x such that Ax  b, x 0.

where From the previous considerations, the corresponding matrix R 2 L R has entries in {0, 1}.

A = 0 B B B B B B B B @ 1 1 1 1 1 . . . . . . 1 1 1 C C C C C C C C A ,b = 0 B B B B B @ 1 0 . . . 0 1 C C C C C A ,c = 0 B B B B B B B B @ p0
(z 0 2p 1 ) 2(z 1 p 2 ) . . . 2(z n 1 p n ) 1 C C C C C C C C A . Now,

Robust Seriation Algorithms

We compare several methods to address the R2SUM( ) problem. First, observe that the objective of R2SUM( ) is not convex. In order to use convex optimization algorithms, it can be relaxed to its convex envelope, resulting in the following problem, minimize

P n i,j=1 A ij h (|⇡ i ⇡ j |) such that ⇡ 2 P.
(HuberSUM( ))

where h (x) is the Huber function, which equals x 2 when |x|  ,a n d (2|x| ) otherwise. In Figure 4.3 are shown plots of the square (`2), absolute value (`1), Huber, and truncated square loss functions appearing in (2-SUM), (1-SUM), (HuberSUM( )), and (R2SUM( )).

QAP solvers (FAQ and PHCD)

The first strategy is to directly minimize the objective of R2SUM( ) using QAP solvers. Indeed, the problem matches QAP(A,B) with B ij =m i n ( , |i j| 2 ). We test the aforementioned For completeness, we also used these QAP solvers in the experiments to solve 2-SUM (i.e.

QAP(A,B) with

B ij = |i j| 2 )
, and HuberSUM( ) (B ij = h (|i j|)).

Symmetry issue in the Permutahedron PH

A typical convex relaxation work-flow involves relaxing both the objective function to its convex envelope, and relaxing the constrained set to its convex hull, in order to use of the arsenal of convex optimization, including scalable first order methods. Here, we seek to optimize the objective functions of 2-SUM and HuberSUM( ), f 2SUM and f Huber , on the convex hull of the set of permutation vectors P n , the polyhedron PH n .

Unfortunately, the solution of a relaxation x 2 PH n does not necessarily (and most of the time, not) lie in P n . To retrieve a solution in P n , one must project the relaxed solution x onto the set of permutations P n , which may be challenging. Here, the flat vector c n , n+1 2 1 n 2 PH n minimizes f 2SUM and f Huber in PH n . Indeed, all its entries being equal, f 2SUM (c n )= f Huber (c n )=0 , which is optimal since these sums involve only non-negative terms. Yet, this optimum is non-informative. Any permutation ⇡ 2 P n has the same distance to c n ,

d = P n i=1 ( n+1 2
i) 2 , thus projecting back c n to P n is completely degenerate. This is illustrated in Figure 4.4, where PH 3 is a salmon-colored hexagone centered around c 3 (red circled dot), and whose vertices are the permutations. PH 3 is represented on a planar figure since PH n lies in a hyperplane of dimension n 1,

H n = {x 2 R n |x T 1 = n(n+1)

2

}. Indeed, all permutation vectors have the same set of elements, hence the same sum, and also the same norm, as one can see from the black dashed circle of fixed norm in Figure 4.4 on which all permutations lie. The symmetry of center c n , formally defined by T n (x) c n = (x c n ), is visible from the level lines of f 2SUM (blue ellipses). The objectives from 2-SUMand HuberSUM( ) are invariant under the "flipping" operator T n . For instance, the permutation ⇡ =(1, 3, 2) T and its symmetric

T 3 (⇡)=( n + 1)1 ⇡ =( 3 , 1, 2)
are on the same level line. This is the fundamental reason why the minimum of 2-SUM and HuberSUM( ) lies in the center, making the basic convex relaxation in PH n useless. . The blue ellipses are the level curves of f 2SUM . The black dashed circle represents the set of points having the same norm as the permutation vectors, and the black diamond is the minimizer of 2-SUM among them. The green (resp. orange) line is where the "good" (resp. "bad") tie-breaking constraint ⇡ 2 +1  ⇡ 3 (resp. ⇡ 1 +1  ⇡ 3 ) is active, and the green (resp. orange) diamond is the minimizer of f 2SUM on the corresponding constrained set, the triangle (2, 1, 3), (1,2,3), (1,3,2) [resp.

(3,1,2) (2,1,3) (1,2,3) (1,3,2) (2,3,1) (3,2,1)
(3, 1, 2), (2, 1, 3), (1,2,3) ]. The closest permutation to the green diamond is (2, 1, 3), which is the correct solution (minimizer of f 2SUM on P 3 ), but the orange diamond is closer to (1, 2, 3) because of the anisotropy induced by the tie-breaking constraint. (2, 1, 3), (1,2,3), (1,3,2) . We consider methods employing both strategies in what follows.

Frank-Wolfe with tie-breaking constraint (FWTB)

The conditional gradient (Frank-Wolfe) Algorithm 4.1 is suited to optimization in PH n since the linear minimization oracle (LMO) performed at each iteration (4.2) boils down to sorting the entries of a vector g 2 R n (hence, it has a computational complexity of O(n log n)). Specifically, the LMO solves, minimize

P n i=1 ⇡ i g i such that ⇡ 2 PH n (LMO)
where g i is the i-th entry of the gradient of the loss function. This linear form is minimized on a vertex of PH, i.e. on a permutation ⇡ ⇤ . Let 2 P n be a permutation that sorts the entries of g by decreasing order, such that g 1 ... g n , then ⇡ ⇤ is defined by ⇡ ⇤ 1 =1,...,⇡ ⇤ n = n. The method (FWTB) adds a tie-breaking constraint (e.g., ⇡ 1 +1 ⇡ n ) in order to break the symmetry and exclude the center c n from the feasible set, as suggested by Lim and Wright Proof. Without loss of generality, let us assume for simplicity that g is already sorted by decreasing value. Let ⇡ ⇤ be the solution of LMO.I f⇡ ⇤ i +1  ⇡ ⇤ j , then ⇡ ⇤ is also solution of LMO-tb. Otherwise, the solution of LMO-tb will be a permutation ⇡ where the constraint is active : ⇡ i +1 = ⇡ j [Lim and Wright, 2014]. Let k = ⇡ i . There are n 1 possible values for k : {1,...,n 1}.F o rag i v e nk, the vector ⇡k , the restriction of ⇡ to the n 2 indexes other than i and j is given by Smith's rule : it is the concatenation of the remaining values ⇡k =(1,...,k 1,k +2,...,n) (given that g is sorted). Therefore, the permutation ⇡ optimal for LMO-tb is Algorithm 4.2 Minimizing g T ⇡ over PH n with tie-break ⇡(i)+1 ⇡(j).

1: g 0 , sort g in decreasing order ( i.e., g( 1 ) ... g( n ) ) ⇡(l) z( 1 (l)) for l 2 {1,...,n} r {i, j} 9:

2: for k 1 to n 1 do 3: if g 0 (k) < g(i)+g(j)
⇡(i) k 10:
⇡(j) k +1. 11: end for Output: A permutation ⇡ (T ) . determined by k. Let us note g 2 R n 2 the vector g without the two entries corresponding to indexes i and j,t h a ti st os a y ,i fi<j, g =( g 1 ,...,g i 1 ,g i+1 ,...,g j 1 ,g j+1 ,...,n).T ok n o w the optimal value of k, let us observe the difference between the objective of LMO-tb for k = K and k = K +1, with 1  K  n 2. For a given k, he objective in LMO-tb can be written as the sum gT ⇡k + kg i +(k + 1)g j . Let us write the tilde scalar product part first.

gT ⇡K =1g 1 +2g 2 + ...+( K 1)g K 1 +( K + 2)g K +( K + 3)g K+1 + ...+ ng n 2 gT ⇡K+1 =1g 1 +2g 2 + ...+( K 1)g K 1 + K gK +( K + 3)g K+1 + ...+ ng n 2
The difference between the objective values for k = K and k = K +1 is therefore ∆ K = 2g K (g i + g j ). Since we assumed g sorted by decreasing order, g also is, and consequently, ∆ K decreases with K.T h e o p t i m a l K ⇤ is therefore the smallest (first) index k for which gk <

(g i +g j ) 2
,a n di fg k

(g i +g j ) 2
for all k 2 {1,...,n 2}, then K ⇤ = n 1.

Graduated

function f 2SUM = x T L A x is replaced by, f (x)=x T L A x µkPxk 2 = x T (L A µP )x,
where L A = diag(A1) A is the Laplacian of A and P = I 1 n 11 T projects on the subspace orthogonal to 1. To keep the problem convex, µ needs to be smaller than 2 , the smallest non-zero eigenvalue of L A . Still, for small values of 2 , this may lead to solutions lying close to the center c up to numerical precision. Also, for f Huber , the convexity is broken for any positive value of µ.

Evangelopoulos et al. [2017a]

proposed a graduated non-convexity scheme called GnCR to solve 2-SUM, where µ is gradually increased in outer iterations of the problem, starting with as m a l lv a l u e( µ  2 ) preserving convexity, and moving towards high values of (µ max ), making the objective concave. This strategy aims at finding a sequence of solutions to the subproblems that follow a path from near c n (when the objective is convex) towards a permutation (when it is concave). To solve each subproblem, GnCR uses the Frank-Wolfe algorithm in PH n without tie-breaking constraint. In Evangelopoulos et al. [2017b], the approach is extended to a pseudo-Huber loss, thus approximately solving HuberSUM( ), with a method called HGnCR.

We include both methods in the experiments.

Unconstrained Optimization in H n with Iterative Bias (UBI)

We propose a method based on unconstrained optimization. We also add a penalty to f Huber in order to avoid the center c and aim to minimize, fHuber (x)=f Huber (x) µh(x), where h is a penalty function pushing away from c. In practice, we use a sigmoidal penalty,

h w (x)= ⇣ 1 + exp hx c, w ci ⌘ 1 .
It breaks the symmetry by adding a bias in a given direction w. However, the penalty h becomes negligible compared to f Huber (x) when kx ck gets large, and the minimizer of fHuber (x) remains bounded. Up to a scaling of µ, it will lie in PH n . Hence, we can use unconstrained optimization to find a minimizer of fHuber in PH n without enforcing the membership to PH n explicitly.

Algorithm 4.3 Iterative scheme with biased unconstrained optimization in H n (UBI).

Input: An objective function f , an initial bias direction ⇡ (1) 2 P n , an increasing bias function h : R ! R, a maximum number of outer iterations T , an optimization algorithm A.

1: for t =1to T do 2: Compute x (t+1) ⇤ 2 argmin x2Hn f (x) h ⇡ (t) (x) using algorithm A.( 4 . 4 ) 3:
Set

⇡ (t+1) = argsort x (t+1) ⇤ 4: end for Output: A permutation ⇡ (T ) .
We propose an iterative method where each outer iteration t solves a sub-problem biased towards the optimum x ⇤ (t 1) found at the previous iteration, described in Algorithm 4.3. The algorithm A used in practice in (4.4) is the LBFGS method, using the implementation from Schmidt [2005]. Figure 4.5 illustrates the iterative procedure. The colored crosses indicate the ⇤ obtained at the outer loops of the algorithm, and the associated colored arrows in the center point towards the associated bias that was used at iteration t.I n blue are the level lines of

(3,1 ,2) (2,1,3) (1,2,3) (1,3,2) (2,3,1) (3,2,1)
n f (x) h x (t) ⇤ (x) o with f = f 2SUM and t =5(red arrow).
minima of the sequence of biased functions. The last one (with the level lines) is biased towards the optimum. Empirically, we found better results by using We could have used h x (t) ⇤ rather than h ⇡ (t) in step 4.4 of Algorithm 4.3, but we empirically found better results with the latter option. Optimization of f in H n is done through unconstrained optimization in R n 1 of the composition of f with an affine transformation described in the following.

We have seen in § 4.3.2 and Figure 4.4 that the set of permutation lie in a hyperplane of

dimension n 1, H n = {x 2 R n |x T 1 = n(n+1) 2
} (which simply means that all permutation vectors have the same sum). Thus, we compute a basis of H n and use an affine transformation from R n 1 to H n such that 0 n 1 corresponds to c n =(n + 1)/21 n 2 H n . In practice, we used, U =(u (1) ,...,u (n 1) ) 2 R n⇥n 1 , e.g., u (j) = ũ(j)

kũ (j) k ,w i t h 8 > > < > > : ũ(j) i =0 if i<j, ũ(j) j = j ũ(j) i =1 if i>j,
The vectors {u (j) } 1jn 1 are orthonormal and are all orthogonal to 1 n .A n yp o i n tx 2 H n can be written as x = A(y) , Uy + c n with y 2 R n 1 .F o r a n y f : R n ! R, we define f Hn : R n 1 ! R by f Hn (y)=f (Uy + c n ) in order to perform unconstrained optimization (UBI) on f Hn . The intersection of the sphere Σ n with H n , represented by the black dashed circle in Figure 4.4, is the transformation of a sphere Σn 1 by A.

Spectral relaxation for HuberSUM(δ)

Let us recall the spectral method introduced in Chapter 1, here in Algorithm 4.4.R o u g h l y ,i t aims to minimize the 2SUM objective defined as,

n X i,j=1 A ij x i x j 2 = x T L A x, (2SUM)
over the set of permutations, by relaxing the integer constraints on permutations, leading to an eigenvalue problem. The spectral method minimizes f 2SUM on a sphere of fixed norm by Algorithm 4.4 Spectral ordering [Atkins et al., 1998]

Input: Connected similarity matrix A 2 R n⇥n 1: Compute Laplacian L A = diag(A1) A 2: Compute second smallest eigenvector of L A , f 1 3: Sort the values of f 1 Output: Permutation : f 1 ( (1))  ...  f 1 ( (n))
computing a second extremal eigenvector, thus resolving the center issue. As we have seen in Section 1.2.2, up to a translation and dilatation of this sphere, it is the sphere with fixed sum (x T 1 = n(n + 1)/2) and fixed norm (kxk 2 2 = n(n + 1)(2n + 1)/6) on which all permutations lie. For n =3, it is represented by the black dashed circle in Figure 4.4.

As we noted earlier, the spectral relaxation relies on the quadratic nature of the 2-SUM objective, and there is no general method for performing convex optimization on a sphere (we can seamlessly deal with linear equality constraints, but not quadratic constraints such as the fixed norm constraints). Optimizing HuberSUM( ) over a sphere is therefore challenging. We propose to extend the spectral Algorithm 1.1 to HuberSUM( ) through the variational form of the Huber loss (so-called ⌘-trick). The absolute value of a real number x 2 R can be expressed as the solution of an minimization problem over a real variable ⌘,

2|x| =min ⌘ 0 x 2 ⌘ + ⌘, (4.5)
and the minimum is realized for ⌘ ⇤ = |x|. Similarly, for any 0, the Huber function defined by

h (x)= 8 < : x 2 if |x|  , (2|x| ) otherwise (4.6)
can be expressed, up to an affine transformation, as the minimum of the same function of ⌘,

but on [ , +1] instead of [0, +1], h (x) ' min ⌘ x 2 ⌘ + ⌘. (4.7)
Specifically, the equation for the Huber function as defined in (4.6)i s ,

h (x)=min ⌘ x 2 ⌘ + ⌘ ! , (4.8)
and the minimum is realized for ⌘ ⇤ = max( , |x|), i.e., ⌘ ⇤ = |x| when |x| ,a n d⌘ ⇤ = otherwise. Using this variational form, we can write HuberSUM( ) as an optimization problem over variables ⇡ and ⌘ 2 S n , minimize

P n i,j=1 A ij ⇣ (⇡ i ⇡ j ) 2 ⌘ ij + ⌘ ij ⌘ such that ⇡ 2 P, ⌘ ij
, for all i, j.

(⌘-HuberSUM)

The objective in ⌘-HuberSUM is jointly convex in (⇡, ⌘) (sum and combination of linear functions with quadratic over linear). The constraint set for ⌘ is convex, and although P is not, it can be relaxed to PH. However we found empirically that an alternate minimization scheme that is not based on convex optimization but rather exploits the efficiency of the spectral algorithm demonstrates good performances. We present it in Algorithm 4.5. We use the spectral Algorithm 4.5 ⌘-Spectral Alternate Minimization Scheme for HuberSUM( ).

Input: As i m i l a r i t ym a t r i xA 2 S + n , a maximum number of iterations T .

1: Set ⌘ (1) = 1 n 1 T n . 2: for t =1to T do 3: Compute ⇡ (t) 2 argmin ⇡2P n X i,j=1 A ij ⇣ (⇡ i ⇡ j ) 2 /⌘ (t) ij + ⌘ (t) ij ⌘ ,
i.e., ⇡ (t) is solution of (2-SUM) for the matrix A ↵ ⌘ where ↵ denotes the Hadamard (entrywise) division.

4:

Compute

⌘ ⇤ 2 argmin ⌘ n X i,j=1 A ij ⇣ (⇡ (t) i ⇡ (t) j ) 2 /⌘ ij + ⌘ ij ⌘ , i.e., ⌘ ⇤ ij max( , |⇡ (t) i ⇡ (t)
j |),f o ra l l(i, j).

5:

Update ⌘ (t+1) ⌘ (t) +(1 )⌘ ⇤ . 6: end for Output: A permutation ⇡ (T ) . Table 4.1: Kendall-⌧ , HuberSUM( ), R2SUM( ), Robust Seriation (with Froebenius norm) scores for the different methods for n = 200, = 20,a n ds/s lim =5. The results are averaged over 100 instances of A 2 M n ( ,s). The first six methods are used with the 2-SUM loss, the six middle ones with the HuberSUM( ) loss, where was chosen following the rule described at the end of §4.2.2, and the two last middle ones with the R2SUM( ) loss. and Some scores are scaled to simplify the table.

Kendall

Frank-Wolfe with Tie-Break (FWTB) is biased

We have not included the results of the FWTB method in the previous section, as it performed poorly. After investigation, we realized that the tie-break constraint actually introduces a bias in the problem, as we explain in the following. Let us focus on the 2-SUM problem. The loss function is homogeneous,

f 2SUM (tx)= X i,j A ij (tx i tx j ) 2 = t 2 X i,j A ij (x i x j ) 2 = t 2 f 2SUM (x).
Similarly, f 1SUM (tx)=tf 1SUM (x) for t>0. Hence, scaling down a given vector x, e.g.,

letting x 1 2
x, reduces the objective function but does not add information about the optimal permutation (the projection on the set of permutations is the same for both vectors). What we are interested in is to find a direction x ⇤ which is optimal compared to other vectors x of same norm. In the original problem over permutations, all permutation vectors have the same norm.

In the spectral relaxations, we optimize over a sphere. However, when we relax to PH,themost prominent descent direction of the function is towards the center. The tie-breaking constraint prevents iterates reaching the center, but it adds a bias in a given direction because not all points saturating the tie-breaking contraint have the same norm nor the same distance to the center. On the set of points in PH n where the tie-break is active, e.g., {x 2 the point c = c + e n 1 n 1 has a squared distance to c and `2 norm : kc ck 2 2 ' 1, kck 2 2 ' n 3 4 , whereas a permutation ⇡ that satisfies the constraints has a distance to c that scales in n 3 and a larger norm : k⇡ ck 2 2 ' n 3 12 , k⇡k 2 2 ' n 3 3 . Therefore, although the direction c may not be optimal for 2-SUM (compared to other vectors of same norm), the minimizer of 2-SUM with tie-break may be closer to the direction of c than to the optimal one. This is what we observe in Figure 4.4 for the bad (orange, top-right triangle) tie-break.

PH n | x 1 +1  x n },
When n becomes large, this may actually lead to numerical precision issues. Indeed, the n 1 first entries of c are equal. When the optimum x ⇤ in the tie-break-constrained PH gets close to c,thev ariationsamongthen 1 first entries of x ⇤ also shrink and the precision required to sort them (in order to project back onto the set of permutations) may become too high.

In Figure 4.4, we also display a good (green, top-left triangle) tie-break. In practice, although there are n 2 non-redundant choices for the indexes i and j constituting a tie-breaking constraint ⇡ i +1  ⇡ j , we can use the solution ⇡ spectr. of the cheap, spectral ordering (Algorithm 3.1) to find a good candidate tie-break. Specifically, chose i 2 argmin ⇡ spectr. and j 2 argmax ⇡ spectr. . The performances of FWTB with the naive (i =1 , j = n) and spectral-initialized tiebreaking strategies are compared to that of the basic spectral Algorithm 3.1 in Table 4.3 (a -I is appended to the algorithm name for the spectral-initialized tie-break results), with the same experimental setup as in Section 4.2 with matrices in M n ( ,s).

We can see that using a default tie-breaking constraint performs very poorly on average.

Using the solution of the spectral algorithm to define the tie-breaking constraint significantly improves the performance compared to using a default tie-break. Still, it does not outperform the spectral algorithm except in a very low noise setting.

E. coli genome reconstruction

We performed experiments with two ONT bacterial data-sets introduced in Chapter 2, including reads sampled from an Escherichia coli genome [Loman et al., 2015], and from an A. baylyi genome [Recanati et al., 2016]. These data-sets are described in Section 2.3.1 and read-length histograms are given in Figure 2.4. Notably, 50% of the reads from the E. coli data-set are larger than 7kbp, whereas it is only the case for 20% of the A. baylyi reads. We used the minimap2 tool [Li, 2018] with default ONT parameters to compute the overlaps between the reads. For each pair of reads for which minimap2 found an overlap, we set the similarity value between those reads as the output (number of matching bases) from minimap2. This process defined a similarity matrix on which we tested our seriation methods. Among the methods that could scale to this size of problem n ⇠ 10 4 , namely, the Frank-Wolfe based relaxations, UBI and ⌘-Spectral, only ⌘-Spectral gave satisfying results, which we report here. We performed a grid search on the threshold to set on the similarity matrix with 24 linearly spaced values varying between the 40% and 80% percentiles of all similarity entries. For each of them, we compute p = from the number of non-zero entries of the matrix as explained in 4.2.2,a n d kept the permutation yielding the best R2SUM( ) score.

For the E. coli data, this method yielded correctly ordered reads, as one can see in Figure ,   with a Kendall-Tau score of 99.5% with the reference ordering obtained by mapping the reads to a reference genome with minimap2. In comparison, the spectral Algorithm 4.4 has a Kendall-Tau score of 32.6%. For the A. baylyi data, however, the method produced an ordering with mis-assemblies, as one can see on Figure , with a Kendall-Tau score of 90.3% (in comparison, Algorithm 4.4 has 41.5%). Here, we only assess the quality of the ordering, but we have seen in Chapter 2 that a correct layout lead to high quality assembly. The mis-ordered points on Figure are not scattered at random. We expect an assembly resulting from this layout to harbour a few large mis-assemblies, i.e., large portions of genome mis-placed or reversed.

Genome assembly using Hi-C data

We evaluate the ⌘-Spectral method on the real Hi-C data-sets introduced in Chapter 3, Section 3.6.4, where the weighted Kendall-Tau score ⌧ to assess fragmented orderings, and the purity index and cluster distance to assess clustering, are defined. Table 4.4 shows the results

for the Plasmodium knowlesi data, for which we have a ground truth chromosome assignment.

Table 4.5 shows the results for the Spodoptera frugiperda data, for which we do not have such a ground truth clustering. We can see that the ⌘-Spectral method improves upon the basic spectral method for the data-sets where the spectral method is already efficient, but that the method introduced in Chapter 3 combined with tSNE performs better on some data-sets harbouring a cluster structure, such as the Plasmodium knowlesi and Sf669 data.

Conclusion

We introduced the Robust Seriation problem, which arises in e.g. de novo genome assembly.

We show that for a class of similarity matrices modeling those observed in genome assembly, the problem of Robust Seriation is equivalent to a modified 2-SUM problem. This modified problem can be relaxed, with an objective function using a Huber loss instead of the squared loss present in 2-SUM. We adapt several relaxations of permutation problems to this 2-SUM problem with Huber loss and also introduce new relaxations, including the ⌘-Spectral method, which is computationally efficient and performs best in our experiments. Notably, it successfully reorders a bacterial genome from third generation sequencing data.

Introduction

The seriation problem has been studied throughout this thesis. As a reminder, it seeks to reorder a set of n elements given only pairwise similarity information. The resulting ordering should lay similar elements nearby. In practice, this translates to properties on the similarity matrix. We recall key definitions related to seriation from Chapter 4.

Definition 5.1.1. We say that A 2 S n is a strong-R-matrix (or strong Robinson matrix) iff it is symmetric and satisfies A ij  A kl for all (i, j, k, l) such that |i j| > |k l|.

Here, S n denotes the set of real symmetric matrices of dimension n. We write R ⇤ n the set of strong-R-matrices of size n,a n dP n the set of permutations of n elements. A permutation can be represented by a vector ⇡ (lower case) or a matrix Π 2 {0, 1} n⇥n (upper case) defined by Π ij =1iff ⇡(i)=j,a n d⇡ = Πg where g =( 1 ,...,n) T . We refer to both representations by P n and may omit the subscript n whenever the dimension is clear from the context. We say that A 2 S n is pre-R ⇤ if there exists a permutation Π 2 P such that the matrix ΠAΠ T (whose entry (i, j) is A ⇡(i),⇡(j) ) is a strong-R-matrix, and the seriation problem seeks to recover this permutation Π, i.e.,s o l v e find

Π 2 P such that ΠAΠ T 2 R ⇤ (Seriation)
in the variable Π 2 P.

Chapter 4 introduced the problem of Robust seriation, which seeks to find the closest pre-R ⇤ matrix to A and reorder it, solving instead

minimize kS ΠAΠ T k such that Π 2 P,S 2 R ⇤ . (Robust Seriation)
where the variable Π 2 P is a permutation matrix, the variable S 2 R ⇤ is a strong-R-matrix, and the norm is typically either the l 1 norm on components or the Froebenius norm.

The main challenge we have been confronted to when trying to perform genome assembly with Seriation is the presence of repeated regions throughout the genome. Let us say that a fragment of DNA is repeated in two separate locations in the genome. When computing pairwise alignments between sequences (to obtain a pairwise similarity), a read encompassing one of the repeated region can seem to overlap with reads encompassing the other.

In the former framework, each read has a distinct identifier, even though it may essentially contain a repeated sequence. Here, we are interested in a different, more complex problem, where we are not able to distinguish duplicates. If an element appears in two copies, we cannot tell whether one or the other copy is similar to another given element. Instead, we can only access a similarity value aggregated over the two duplicates (this will be formalized in Section 5.2.2).

The chapter is organized as follows. In Section 5.2, we first motivate the problem with the application of cancer genome assembly through Hi-C data. Then, we formalize the problem, starting with an illustrative example. In Section 5.3, we propose an alternate minimization method to solve the problem of seriation with duplications. It involves solving robust seriation, for which we have proposed several algorithms in Chapter 4. Finally, we present numerical results on synthetic data in Section 5.4.

Seriation with Duplications

The reformulation of de novo sequencing as a (robust) seriation problem is based on the assumption that, up to noise, the bins can be reordered to form a long chain. While this hypothesis is relevant when a normal genome or chromosome is sequenced with long reads, it clearly fails to hold in an important case: cancer genomes. Indeed cancer cells typically harbour so-called structural variations where large portions of the genome, up to whole chromosomes, are duplicated or deleted, and where new chromosomes are formed by fusing two pieces of chromosomes which are not connected in a normal genome. For example, Figure 5.1 shows the 1D structure of a breast cancer cell line. Different colors correspond to DNA fragments normally in different chromosomes. Instead of 23 pairs of chromosomes with each pair in a single uniform color, expected in a normal cell, we observe various mosaics of colors indicating various duplication and fusion events. interaction matrix A such that A ij is the frequency of interactions between DNA fragments in bins i and j. Interestingly, most 3D interactions take place between DNA fragments which are on the same chromosome, and the frequency of 3D interactions tends to decrease with the distance between the fragments when they are on the same chromosome; hence Hi-C data can be used to perform genome assembly, using e.g., a seriation algorithm to obtain the layout [START_REF] Korbel | Genome assembly and haplotyping with Hi-C[END_REF]].

Hi-C data

An Hi-C experiment roughly proceeds as follows. Freeze the DNA in its current 3D conformation, and collect pairs of DNA fragments that lie close to each other in this spatial conformation. For every such pair (k, l), map each of the two fragments to a normal reference genome, providing their positions, p k and p l . Add +1 to the interaction matrix entry A ij corresponding to the two bins i and j that respectively span p k and p l . This process is repeated to statistically obtain an average proximity (frequency) between two bins.

Because of duplications, deletions and translocations in cancer genome, each bin (defined according to a normal reference genome) may be included in several fragments of different chromosomes in a cancer genome, and it may therefore not be possible nor relevant to order the bins. Instead, since it is possible to estimate from Hi-C data the total number of DNA copies for each bin, it makes more sense to first associate to each bin a corresponding number of fragments (e.g. two fragments per bin in a normal diploid genome), and then reconstruct an ordering of fragments into a number of chains to estimate the 1D structure of a cancer genome (Figure 5.1).

The difficulty to apply a seriation algorithm is that Hi-C data provide cumulative information at the bin level, not at the fragment level. More precisely, if we denote S kl the (unobserved)

Problem setting

For clarity, let us begin by an example with n =3 , N =4 . Consider a simplified reference genome split in 3 subsequences, g =(♥, ♦, |). In a cancer genome, the ♥ sequence is duplicated and also appears at the end of the genome. Using the symbol ~to denote the duplicated sequence of DNA, the cancer genome can be written g =( ♥, ♦, |, ~). The true interaction matrix between the fragments (♥, ♦, |, ~) is a L R matrix,

S ⇤ = ♥♦|0 B B B @ 1 C C C A ♥ 3210 ♦ 2321 | 1232

~0123

Yet, interactions between (|, ~) and (|, ♥) are both attributed to (|, ♥) by the Hi-C experiment, resulting in the following observed interaction matrix and duplication count vector,

A = ♥♦| 0 B @ 1 C A ♥ 633 ♦ 332 | 323 ,c =(2, 1, 1) T .
Observing A, the sequence we wish to reconstruct is in fact ⇡ ⇤ =(1, 2, 3, 1) T .

Given a matrix A 2 S n of similarity between n bins, and a vector c 2 N n (the "counts" of the bins), with total N = P n i=1 c i , Seriation with Duplications aims at finding a sequence ⇡ 2 [1,n] N of N integers such that i appears c i times in ⇡,a tp o s i t i o n sL i ⇢ [1,N] with

|L i | = c i ,a n dam a t r i xS 2 R ⇤ N such that A ij = X k2L i ,l2L j
S kl for all i, j 2 [1,n].

Remark that if c = 1 n (the vector of R n with all entries equal to 1), the problem is equivalent to seriation and ⇡ is a permutation vector.

To represent the subsets {L i } i2 [1,n] , we use assignment matrices Z 2 {0, 1} n⇥N such that where Z 0 is an initial assignment matrix. Like Seriation, SD may not be feasible. The analog of Robust Seriation is then written

Z ik =1iff k 2 L i (
minimize kZ 0 ΠSΠ T Z T 0 Ak such that Π 2 P N ,S2 R ⇤ N .
(RSD)

Note again that if c = 1 n , then N = n, Z 0 = I n ,a n dSD (respectively RSD) is equivalent to Seriation (resp. Robust Seriation).

Algorithms

Alternate projection for Seriation with Duplications

Let us assume that we are able to project on the set of pre-strong-R matrices, that is to say, given S, we can compute the couple (Π ⇤ ,S ⇤ ) 2 P ⇥ R ⇤ that minimizes kΠRΠ T Sk (note that the projection on the set of pre-strong-R matrices is nothing but the Robust Seriation problem). We can then use alternationg projections to optimize (RSD) (although the set of pre-strong-R matrices is not convex, so convergence to a global optimum is not garanteed). We detail this method in Algorithm 5.1.

In fact, we can use any method presented in the previous chapter (Section 4.2)t os o l v et h e projection step 3 in Algorithm 5.1. In our experiments here, we use ⌘-spectral and UBI, which are the most efficient, and spectral as a baseline. From the permutation Π ⇤ obtained by, e.g., solving HuberSUM( ) with ⌘-Spectral, we compute S ⇤ by doing a `1 projection of Π ⇤ S (t) Π T ⇤ onto R ⇤ through linear programming. Indeed, the membership to R ⇤ can be described by a set of linear inequalities. We can also add upper bounds on the matrix entries belonging to a given diagonal, if we have ap r i o r iknowledge on the law by which the entries decrease when moving away from the diagonal, which is the case for Hi-C genome reconstruction. We detail these steps in Section 5. Input: Am a t r i xA 2 S n , a duplication count vector c 2 N n , a maximum number of iterations T . 1: Set N = P n i=1 c i ,Z (0) 2 Z c and S (0) = Z (0)T diag(c 1 )A diag(c 1 ) T Z (0) , i.e., S

A ij c i c j with k 2 L i and l 2 L j .

2: while t  T do 3:

Compute (Π ⇤ ,S ⇤ ),s o l u t i o no f( Robust Seriation) for S (t) , and set

S (t+ 1 2 ) S ⇤ Z (t+1) Z (t) Π ⇤ 4:
Compute S A , projection of S (t+ 1 2 ) on the set of matrices that satisfy Z (t+1) SZ (t+1)T = A, and set S (t+1) S A 5:

t t +1. 6: if Z (t+1) = Z (t) then 7: break 8:
end if 9: end while Output: Am a t r i xS (T ) , an assignment matrix Z (T )

Algorithms for Robust Seriation

We have studied the (Robust Seriation) problem in Chapter 4 and evaluated various methods designed to solve it. One of the steps of Algorithm 5.1 coincides with (Robust Seriation). In this chapter, we will retain the three following methods evaluated in Chapter 4,

• Spectral (baseline method, Algorithm 4.4)

• ⌘-Spectral (Algorithm 4.5)

• Unconstrained minimization in PH (UBI, Algorithm 4.3), and refer the reader to Section 4.3 for details on these methods.

Algorithmic details

We now detail algorithmic solutions to several subproblems required by seriation with duplications.

Projection on R ⇤ (step 3 of Algorithm 5.1)

In step 3 of Algorithm 5.1, we wish to compute (Π ⇤ ,S ⇤ ),s o l u t i o no f( Robust Seriation) for S (t) . To do so, we can use one of the algorithms presented in Section 4.2. However, these algorithms do not address the problem of Robust Seriation directly. Rather, they seek to find a permutation that is optimal for a objective function which coincides with Robust Seriation for the specific class of M n ( ,s) matrices. Two problems arise then. First, in our Seriation X,

minimize P N k,l=1 |S kl X kl | such that ZXZ T = A.
(dupli-proj)

The constraints impose that for each pair (i, j) 2 [1,n] ⇥ [1,n], A ij = P k2L i ,l2L j X kl , where L i ⇢ [1,N] is the set of indexes assigned to i through the assignment matrix Z. The objective is also separable, since

N X k,l=1 |S kl X kl | = n X i,j=1 X k2L i ,l2L j |S kl X kl |
We can then solve separately, for each pair (i, j), the subproblem, minimize

P k2L i ,l2L j |S kl X kl | such that A ij = P k2L i ,l2L j X kl .
(dupli-proj(i,j))

For a given pair (i, j), L i and L j are known (through Z), and if we consider the vectorization (stacking of the columns into a single vector) of the submatrices X L i ,L j and S L i ,L j , denoted x and s respectively, and denote a = A ij , the subproblem on the variable x reads minimize ks xk

such that x T 1 = a, x 0.
(dupli-proj(i,j))

We impose non-negativity of the coefficients of X since this is part of the definition of similarity matrices. The above general problem of approximating a vector with a non-negative vector of fixed norm can be solved exactly when the norm is the `2 norm (this solution is optimal for the `1 norm too) with Algorithm 5.2.

Numerical Results

We performed synthetic experiments in which we generate the data as follows. We first build as t r o n g -Rm a t r i xS of size N , and a random duplication count vector c 2 N n such that

N = P n i=1 c i .
We generate a random assignment matrix Z 2 Z c , and the corresponding observed matrix A = ZSZ T . We then test Algorithm 5.1 by providing it with A and c and comparing its output Z out and S out to the ground truth. Specifically, we compute the relative Froebenius distance between S and S out ,d 2R = kS S out k F /kSk F , and we compute a distance between the assignment matrices as follows. For a given bin index i 2 [1,n] (i.e. ar o wZ i ), there are c i locations for the non-zeros of the i-th row Algorithm 5.2 Minimizing ks xk with non-negativity (x 0)andsum(x T 1 = a) constraints.

Input: A target vector s 2 R p + ,av a l u ea 0. 1: s 0 , sort s in decreasing order ( i.e., s( 1 ) ... s( n ) )

2: for k 1 to n do 3:

x0 (k) s 0 (k)+ 1 k (a P k i=1 s 0 (i)) 4:
if x0 (k) < 0 then end if 8: end for 9: x 0 (j)=s 0 (j)+ 1 k (a P k i=1 s 0 (i)) for j =1,...,k 10: x 0 (j)=0for j>k 11: x( j )=x 0 (j) for j =1,...,p.

Output: A vector x 2 R p + .
of Z and of Z out (which can also be viewed as two subsets L i and L out i of [1,N]). To compute the distance between these positions, we first compute a matching between the elements of L i and L out i using the Hungarian algorithm [Kuhn, 1955]. Then, we compute the distance between each matched pair of elements (k, k out ) 2 L i ⇥ L out i , and store the average distance between matching pairs for row i. Supplementary Figures D.5 and D.6 illustrates this process. The average over all rows of this average distance is given in Table 5.1 as meanDist, and we also provide its standard deviation and median.

In the experiments, we built dense strong-R, Toeplitz matrices S where the entries follow a The ⌘-Spectral method works best for dense matrices, and is outperformed by H-UBI for maatrices in M N ( ,s). We observe that, as expected, the recovered assignment Z out is closer to Z when N/n is smaller. However, the D2S scores and the qualitative Figures D.1 and D.2 suggest that for large N/n, the recovered matrix S out may be close to S although the assignment is not well recovered. Intuitively, this means the problem is degenerate, with several assignment matrices roughly leading to the same matrix S, and the algorithm finds one of these. = n/5, s =0 , and various values of N/n, where the (Robust Seriation) problem is tackled with either Spectral, ⌘-Spectral or H-UBI within Algorithm 5.1.F r o mt h eo u t p u tS out and Z out of Algorithm 5.1 and the ground truth S and Z from which the data A is generated, D2S is the relative Froebenius distance between S and S out , Huber is the (HuberSUM( ))l o s so nS, meanDist, stdDist and medianDist are the average, standard deviation and median of the distance between the positions assigned to a index k by Z and Z out (see main text for details). Time is the amount of CPU time elapsed until convergence of Algorithm 5.1.

Multiple chromosomes : Seriation+Clustering with Duplications

In the application motivating this problem, the cancer genome to be reconstructed has multiple chromosomes (which may harbour structural variants). In a Hi-C experiment, the interchromosome frequencies of interaction are significantly lower than the intra-chromosome frequencies [Lieberman- [START_REF] Lieberman-Aiden | Comprehensive mapping of long-range interactions reveals folding principles of the human genome[END_REF]. Thus, if reordered correctly, the target similarity matrix S has a block-structure, and each block has a Robinsonian structure. Still, the methods we use for Seriation, and in particular the spectral methods, are not necessarily suited to reordering clustered (block) matrices. We therefore propose to add a clustering step in Algorithm 5.1 in order to leverage the cluster structure. This is summarized in Algorithm 5.3, where we project the current matrix S on the set of block matrices in line 4, and we reorder each cluster in line 5. The projection on block matrices is not exactly a clustering procedure, since we only want to find breakpoints between clusters, but two points can be in the same cluster only if they are contiguous in the current ordering Π ⇤ . To find these breakpoints, we use an algorithm from [START_REF] Ding | Linearized cluster assignment via spectral ordering[END_REF] which seeks the minima of a measure called cluster crossing.

For each point i, it is roughly defined as the sum along the anti-diagonal i in a bandwidth m, where m can be chosen according to the number of target clusters. If the similarity matrix has a cluster structure, then the cluster crossing ⇢ should have local minima at the boundaries between the clusters, as we can see in Figure 5.2b.

⇢(i)= m X j=1 A i j,i+j , (5.1)

Numerical experiments with block + Robinson matrices

We conducted experiments similar to those of Section 5.4 where we start with matrices S that are the sum of a dense Robinson matrix and a block matrix, as the one displayed in Figure 5.2. Input: Am a t r i xA 2 S n , a duplication count vector c 2 N n , a maximum number of iterations T . 1: Set N = P n i=1 c i ,Z (0) 2 Z c and S (0) = Z (0)T diag(c 1 )A diag(c 1 ) T Z (0) , i.e., S

A ij c i c j with k 2 L i and l 2 L j .

2: while t  T do 3:

Compute (Π ⇤ ,S ⇤ ),s o l u t i o no f( Robust Seriation) for S (t) .

4:

Compute S Clus. , projection of S ⇤ on the set of block matrices.

5:

Reorder each block with (Robust Seriation), and update Π ⇤ accordingly.

6:

Set S (t+ 1 2 )

S ⇤ ,a n dZ (t+1) Z (t) Π ⇤ .

7:

Compute S A , projection of S Clus. on the set of matrices that satisfy Z (t+1) SZ (t+1)T = A, and set S (t+1) S A 8: t t +1.

9:

if Z (t+1) = Z (t) then 10: break 11:

end if 12: end while Output: Am a t r i xS (T ) , an assignment matrix Z (T ) In Table 5.3, we provide the results of such experiments with 5 clusters, where we use the ⌘-Spectral algorithm to solve (Robust Seriation)i nA l g o r i t h m s5.1 and 5.3.T a b l eD.6 is the analog with 10 clusters, for which the results are similar. We observe that when N/n is suffiently small (N/n =1 .33), the clustering step clearly helps the algorithm to converge to a good minimum. However, the performance gain becomes marginal and both Algorithms 5.1 and 5.3 perform poorly with higher values of N/n.

Discussion

We have introduced the problem of Seriation with Duplications. It generalizes Seriation and can adapt the problem of performing genome assembly from Hi-C frequency matrices to genomes with structural variants. After formalizing the problem setting, we present an algorithmic scheme based on seriation and alternate projections between the set of duplications constraints and the set of Robinson matrices. We evaluate this method synthetic experiments on stylized matrices modeling Hi-C experiments from single-chromosome genomes, with duplications.

Then, we outline the issues due to the presence of multiple chromosomes. We propose to modify the alternate projections algorithm to handle the cluster structure, and evaluate it on synthetic data, on which it perform moderately well. Conclusion and Perspectives

Summary of the thesis

Throughout this thesis, we have studied the problem of seriation with the aim of solving an important genome assembly task. We explored existing techniques and develop new methods in order to meet the challenges arising with real genomic data.

In the opening chapter, we formalized the seriation problem mathematically. We recalled algorithmic challenges and detailed existing approaches constituting the basis of our work.

Then, we presented genome assembly techniques, highlighted the key challenges, and reviewed major sequencing technologies.

In our first contribution, we made the proof of concept that the seriation framework was suited to de novo genome assembly. To this end, we applied a standard spectral, seriation algorithm to real de novo assembly problems, using third-generation sequencing data. We integrated our seriation module seamlessly in an end-to-end overlap-layout-consensus assembly scheme. This yielded competitive experimental results compared to state-of-the-art methods, although it was challenged by the repeated regions occurring in DNA sequences, leading to possibly fragmented assemblies.

In our second contribution, we borrowed from spectral graph theory and embedding techniques to propose an extension of the spectral method used in the first contribution (Chapter 2).

It provided a unifying framework for seriation and circular seriation, a variant of the problem where instead of seeking for an ordering of data along a linear chain, we search for a cyclic ordering of the data, where the objects at the end of the cycle are similar to those at the beginning of it. We derived theoretical guarantees for circular seriation analog to those existing for linear seriation. Notably, our extended spectral method significantly improves the robustness of the original spectral method when the data is corrupted by noise. We evaluated it on several types of data, including third-generation sequencing data for de novo assembly, spatial conformation (Hi-C) data, and single-cell Hi-C data used in a cell-cycle ordering problem.

In our third contribution, we attempted to model the problem of performing seriation on data corrupted by noise, and introduced the framework of robust seriation. We showed how several seriation-like optimization problems relate for stylized matrices modeling those observed in de novo assembly (including the noise). Then, we explored several algorithmic approaches, including recently developed methods for permutation problems, and new methods that we introduced, to tackle these problems. We compared experimentally this set of methods on synthetic and real sequencing data. Some of our methods are substantially more robust to noise than the basic spectral method, but the de novo assembly experiments do not support improvement over the spectral extension presented in Chapter 3.

Finally, our last contribution was to introduce the problem of seriation with duplications.

It is an extension of the seriation problem, motivated by the assembly of cancer genomes from Hi-C data. We described how structural variations arising in cancer genomes lead to Hi-C frequency matrices which cannot be used for assembly in a standard seriation framework. We then formalized the problem setting, and proposed an alternate projection scheme to tackle it. We evaluated this method on synthetic data modeling a single-chromosome genome with structural variations. Then, we attempted to adapt the method to genomes with multiple chromosomes, and performed additional synthetic experiments.

Perspectives

The work presented in this manuscript calls for subsequent development, in both applications and theory. Let us list key future work directions.

i Integration of new seriation methods in a full-assembly pipeline. In Chapters 3 and 4,w e introduced new algorithmic methods for the seriation problem with enhanced robustness, and evaluated them for the layout computation in de novo assembly experiments. However, we did not integrate them into the full assembly pipeline presented in Chapter 2. Thus, when testing the methods, we only evaluated ordering produced, but not the eventual output DNA assembly. Since our new methods produced accurately ordered layouts with fewer contigs than the spectral method used in Chapter 2, we can expect the resulting assembly to be of better quality, with a smaller number of mis-assemblies. Still, it would be interesting to quantify how improvements on the layout translate to the consensus produced.

ii Extensions of Chapter 3. In Chapter 3, we derived theoretical guarantees for circular seriation, but with more restrictive assumptions than those used for linear seriation in Atkins iii 10X Genomics data. We have mostly dealt with third-generation sequencing data in our experiments. Such data contains long-read (tens of kbp), allowing for an easier resolution of the repeats than with short-reads, where using additional pair-end information is necessary to construct the layout through a scaffolding procedure. A recent development in sequencing technology commercialized by 10X genomics combines short-reads with so-called molecular barcoding, linking short-reads to long molecules to provide long-range information. In short, a barcode is associated to regions of DNA of large length (a few tens of kbp), and two reads that are close to each other on a DNA strand are likely to share several barcodes. This barcoding notably permits to call structural variants and distinguish between haplotypes for diploid genomes (such as the human genome). The Supernova assembler [Weisenfeld et al., 2017] is based on a short-reads assembly scheme, and the additional molecular barcode information is used to disambiguate the scaffolding. It would be of major interest to see whether the barcode information is sufficient to find the layout of the short-reads with seriation, without even computing overlaps. Specifically, we could define a pairwise similarity between reads as the number of barcodes they share, and apply a clustering/reordering method directly on it.

iv Algorithms to perform Clustering+Seriation. In Chapter 2, we attempted to assemble eukaryotic genomes with seriation methods. These genomes contain multiple chromosomes.

Though, due to repeats occurring in distinct chromosomes, the read-overlap based similarity matrices contain non-zero values between reads coming from distinct chromosomes. We have shown through experiments related to optical mapping that having prior information about the chromosome assignment of the reads (given a read, know to which chromosome it belongs) improved the quality of the assembly. In practice, we do not have such information, and our methods need to fragment the assembly into many contigs to avoid mis-assemblies where contigs from distinct chromosomes are mixed together. Then, in Chapter 3, we conducted experiments with Hi-C frequency matrices having a block structure corresponding to distinct chromosomes. Although the inter-chromosomes similarity is generally smaller than the intra-chromosome similarity for Hi-C data, there are still some high similarity values between chromosomes. On synthetic data where the cluster structure is prominent, our method can naturally split the data in sub-orderings. Indeed, Algorithm 3. v Seriation with duplication on real Hi-C data from cancer genomes. In Chapter 5,w ei n t r oduce the problem of seriation with duplications in order to assemble genomes with structural variants from Hi-C data. Yet, we only test our method on synthetic data. A key issue arising with real data is related to the previous point, namely that the genome is divided into distinct chromosomes. Still, in seriation with duplications, there is an additional level of complexity compared to regular seriation, for the clustering step also. Indeed, the clusterstructure appears on the hidden similarity matrix S, but not on the observed, cumulative matrix A. In Chapter 5, we consider adding a clustering step to the alternate projections scheme. However, this only enhances the results with few duplications, on synthetic, wellconditioned matrices that are the sum of a block matrix and a Robinson matrix. Regarding the previous item from this list, finding a principled method solving Clustering+Seriation could be used in the alternated projection scheme instead of regular seriation. Also, another issue with our proposed method is its algorithmic complexity. The projection on the set of R-matrices is done through a linear program and do not scale to large similarity matrices.

Finally, Hi-C data from cancer genomes have additional structure that is not leveraged here.

The structural variants consist of entire blocks of DNA that are duplicated and merged.

Within a block, the ordering is the same as for the reference genome. Taking this structure into account would likely improve the method. Indeed, the Racon pipeline maps the reads to a draft sequence to get the layout and then computes consensus sequences in windows across the genome. Our pipeline instead directly computes the layout and then generates consensus sequences in windows across the genome (the latter task being embarassingly parallel). Canu is faster than our method on the PacBio datasets (probably at least because because we did not adapt our pipeline (as Canu does) to the much higher coverage, nor to the higher fraction of chimeric reads typical of PacBio data), but not on the ONT datasets. The memory for the spectral method can be allocated among several cores.

A.1.2 Runtime for layout only

The running time of the sole spectral method in Table A.2 provides the assembly results obtained by using the reads corrected by Canu's correction module. These corrected datasets were obtained by running Canu with the saveReadCorrections=True option on the datasets presented in 2.3.1. Canu includes correction and trimming, resulting in a removal of short reads and a lower coverage than in the original raw data. However, it is the coverage of the raw dataset which is relevant since higher coverage in the latter will result in longer reads in the corrected data, even though the coverage in all corrected datasets are roughly below 40x. We indicate the coverage of the corrected datasets in parentheses next to the coverage of the original dataset. For the spectral method, we give the results after the contig merging step (see 2.3.3). The number of contigs before this post-processing is given between parentheses. Unlike with raw data, the polishing effect of adding Racon to our pipeline is not significant. All methods have comparable results on the corrected datasets. The best result in terms of average identity only is indicated in bold (but other metrics should also be used to compare the assemblies).

Table A.3 is a mis-assembly report obtained with QUAST [Gurevich et al., 2013]( o n l ya subset of the report is shown). Given the accuracy of the Miniasm assembly, it is likely that the zeros in the Miniasm column are due to the fact that the algorithm failed to correctly match the sequences, rather than the absence of misassemblies. On all ONT datasets, the Spectral and Spectral+Racon methods are among those yielding the least global misassemblies (relocation, translocation or inversions). A.3 for the corrected datasets. We observe that the number of local misassemblies is smaller than with the uncorrected data, but the number of global ones is not. None of the assemblers has a significantly smaller or larger number of misassemblies compared to the others.

Table A.4 is the analog of Table

A.5 Implementation and reproducibility

Spectrassembler is implemented in python and available on https://github.com/antrec/ spectrassembler with a usage example of how to reproduce the results obtained with E. coli ONT data. We used the following software :

• SPOAhttps://github.com/rvaser/spoa we check whether the elements in one of the two ends of c i have edges with those in one of the two ends of c j in the graph defined by A. According to that measure of similarity and to the direction of these meta-edges (i.e., whether it is the beginning or the end of c i and c j that are similar), we merge together the two subsequences that are the closest to each other.

We repeat this operation with the rest of the subsequences and the sequence formed by the latter merge step, until there is only one final sequence, or until the meta similarity between subsequences is zero everywhere. We formalize this procedure in the greedy Algorithm B.1, which is implemented in the package at https://github.com/antrec/mdso.

Given C reordered subsequences (one per connected component of S) (c i ) i=1,...,C , that form ap a r t i t i o no f{1,...,n}, and a window size h that define the length of the ends we consider (h must be smaller than half the smallest subsequence), we denote by c i (resp.

c + i ) the first (resp. the last) h elements of c i ,a n da(c ✏ i ,c ✏ 0 j )= P u2c ✏ i ,v2c ✏ 0 j
A uv is the similarity between the ends c ✏ i and c ✏ 0 j , for any pair c i ,c j , i 6 = j 2 {1,...,C}, and any combination of ends ✏, ✏ 0 2 {+, }. Also, we define the meta-similarity between c i and c j by,

s(c i ,c j ) , max(a(c + i ,c + j ),a(c + i ,c j ),a(c i ,c + j ),a(c i ,c j )) , (B.1)
and (✏ i , ✏ j ) 2 {+, } 2 the combination of signs where the argmax is realized, i.e.,s u c ht h a t

s(c i ,c j )=a(c ✏ i i ,c ✏ j j )
. Finally, we will use ci to denote the ordered subsequence c i read from the end to the beginning, for instance if c =(1,...,n), then c =(n, . . . , 1).

Algorithm B.1 Merging connected components

Input: C ordered subsequences forming a partition P =( c 1 ,...,c C ) of {1,...,n},a ni n i t i a l similarity matrix A, a neighborhood parameter h. Compute meta-similarity S such that Sij = s(c i ,c j ), and meta-orientation (✏ i , ✏ j ), for all pairs of subsequences with equation B.1.

3:

if S =0then find (i, j) 2 argmax S,a n d(✏ i , ✏ j ) the corresponding orientations.

7:

if (✏ i , ✏ j )=(+, ) then 8:

c new (c i ,c j ) 9:
else if (✏ i , ✏ j )=(+, +) then 10:

c new (c i , cj ) 11: else if (✏ i , ✏ j )=( , )) then 12: c new (c i ,c j ) 13: else if (✏ i , ✏ j )=( , +)) then 14: c new (c i , cj ) 15:
end if

16:

Remove c i and c j from P .

17:

Add c new to P .

B.1.2 Computing Kendall-Tau score between two permutations describing a circular ordering

Suppose we have data having a circular structure, i.e.,w eh a v en items that can be laid on a circle such that the higher the similarity between two elements is, the closer they are on the circle. Then, given an ordering of the points that respects this circular structure (i.

KT(i)

Kendall-Tau( , ⇡(i), ⇡(i + 1),...,⇡(n), ⇡(1),...,⇡(i 1) )

B.2 Additional Numerical Results

B.2.1 Genome assembly experiment (detailed)

Here we provide details about the application of seriation methods for genome assembly and details about our experiment. We used the E. coli reads from Loman et al. [2015]. They were sequenced with Oxford Nanopore Technology (ONT) MinION device. The sequencing experiment is detailed in http://lab.loman.net/2015/09/24/first-sqk-map-006-experiment where the data is available. We also used the A. baylyi dataset sequenced at the Genoscope, introduced in Chapter 2, Section 2.3.1. The overlaps between raw reads were computed with minimap2 [Li, 2018] with the ONT preset (minimap2 -x ava ont). The similarity matrix was constructed directly from the output of minimap2. For each pair (i, j) of reads where an overlap was found, we let the number of matching bases be the similarity value associated (and zero where no overlap are found). The only preprocessing on the matrix is that we set a threshold to remove short overlaps. In practice we set the threshold to the median of the similarity values, i.e., we discard the lower half of the overlaps. We then apply our method to the similarity matrix. The laplacian embedding is shown in Figure B.2a. We used no scaling of the Laplacian as it corrupted the filamentary structure of the embedding, but we normalized the similarity matrix beforehand with W D 1 WD 1 as in Coifman and Lafon [2006]. The resulting similarity matrix S computed from the embedding in Algorithm 3.3 is disconnected. Then, Algorithm 3.1 is applied in each connected component, yielding a fragmented assembly with correctly ordered contigs, as shown in Figure B.2b. However, if the new similarity matrix S is disconnected, the input matrix A is connected. The fragmentation happened while "scanning" the nearest-neighbors from the embedding. One can therefore merge the ordered contigs using the input matrix A as follows. For each contig, we check from A if there are non-zero overlaps between reads at the edges of that contig and some reads at the edges of another contig. If so, we merge the two contigs, and repeat the procedure until there is only one contig left (or until there is no more overlaps between edges from any two contigs). This procedure is detailed in Algorithm B.1.N o t et h a tt h eE. coli genome is circular, therefore computing the layout should be casted as a Circular Seriation problem, as illustrated in Figure 3.1. Yet, since the genome is fragmented in subsequences since S is disconnected, we end up using Algorithm 3.1 in each connected component, i.e., solving an instance of Linear Seriation in each contig.

The experiment can be reproduced with the material on https://github.com/antrec/ mdso, and the parameters easily varied. Overall, the final ordering found is correct when the threshold on the overlap-based similarity is sufficient (in practice, above ⇠ 50% of the nonzero values for E. coli,a n d⇠ 70% for A. baylyi). When the threshold increases or when the number of nearest neighbors k from Algorithm 3.3 decreases, the new similarity matrix S gets more fragmented, but the final ordering remains the same after the merging procedure, except for very large values where many reads end up with no overlap with any other read. A good heuristic to choose the threshold value is to take the highest value that leaves the resulting merged component contiguous (and of size comparable to the number of input reads, meaning that few reads lost all their overlaps/edges in the thresholding procedure).

B.2.2 Gain over baseline

In Figure 3.5, each curve is the mean of the Kendall-tau (a score directly interpretable by practitioners) over many different Gaussian random realizations of the noise. The shaded confidence interval represents the area in which the true expectation is to be with high probability but not the area in which the score of an experiment with a given noisy similarity would be. As mentioned in the main text, the shaded interval is the standard deviation divided by p n exps , since otherwise the plot was hard to read, as the intervals crossed each others.

Practitioners may use this method in one-shot (e.g. for one particular data-set). In that case, it would be more relevant to show directly the standard deviation on the plots, which is the same as what is displayed, but multiplied by 10. Then, the confidence intervals between the baseline and our method would cross each other. However, the standard deviation on all experiments is due to the fact that some instances are more difficult to solve than some others. In this Section, we prove Theorem 3.3.2. There are many technical details, notably the distinction between the cases n even and odd. The key idea is to compare the sums involved in the eigenvalues of the circulant matrices A 2 C ⇤ R .I ti st h es u mo ft h eb k times values of cosines. For 1 , we roughly have a reordering inequality where the ordering of the b k matches those of the cosines. For the following eigenvalues, the set of values taken by the cosines is roughly the same, but it does not match the ordering of the b k . Finally, the eigenvectors of the Laplacian of A are the same than those of A for circulant matrices A, as observed in §3.3.3.

B.2.6 Supplementary Figures for Hi-

We now introduce a few lemmas that will be useful in the proof.

Notation. In the following we denote z ⇧ . Also when m and n are not coprime we will note m = dm 0 as well as n = dn 0 with n 0 and m 0 coprime.

B.3.1 Properties of sum of cosines.

The following lemma gives us how the partial sum sequence (S

q ) behave for q = p or q = p 1 as well as it proves its symmetric behavior in (B.3). Also, for 1  q  p/2, S

S

q .

(B.3)

For n and m 2 even (n =2p), we have S

p 1 q = S (1)

q for 1  q  (p 1)/2 (B.4) S (1)
p 1 =0 and S Proof.

Let us derive a closed form expression for the cumulative sum S (m) q , for any m, q 2 {1,...,p}

S (m) q = P q k=1 z (m) k =R e ⇣ P q k=1 e 2i⇡km n ⌘ =R e ⇣ e 2i⇡m/n 1 e 2i⇡qm/n
1 e 2i⇡m/n ⌘ = cos ⇡(q + 1)m/n sin(⇡qm/n) sin(⇡m/n) .

(B.7)

Let us prove equation (B.2) with the latter expression for q = p. Given that n =2p +1= 2(p +1/2), we have,

⇡(p + 1)m n = ⇡(p +1/2+1/2)m 2(p +1/2) = ⇡m 2 + ⇡m 2n , ⇡pm n = ⇡(p +1/2 1/2)m 2(p +1/2) = ⇡m 2 ⇡m 2n .
Now, by trigonometric formulas, we have, Let us now prove equation (B.3) with a similar derivation. Let f (q) , cos ⇡(q+1)/n sin(⇡q/n), defined for any real q 2 [1,p/2]. We wish to prove f (p q) f (q) for any integer q 2 {1,...,bp/2c}.U s i n gn = 2(p +1/2), we have,

cos ✓ ⇡m 2 + x ◆ = 8 > < > : ( 1) m/2 cos (x), if m is even ( 1) (m+1)/2 sin (x), if m is odd sin ✓ ⇡m 2 x ◆ = 8 > < > : ( 1) (1+m/2) sin (x), if m is even (
⇡(p q + 1) n = ⇡(p +1/2 (q 1/2)) 2(p +1/2) = ⇡ 2 ⇡(q 1/2) n , ⇡(p q) n = ⇡(p +1/2 (q +1/2)) 2(p +1/2) = ⇡ 2
⇡(q +1/2) n .

then (B.12) holds strictly.

The following proposition gives the usual derivations of eigenvalues in the R-circular Toeplitz The following proposition is a crucial property of the eigenvalues of a circular Toeplitz matrix. It later ensures that when choosing the second eigenvalues of the laplacian, it will corresponds to the eigenvectors with the lowest period. It is paramount to prove that the latent ordering of the data can be recovered from the curve-like embedding.

Proposition B.3.5. A circular-R, circulant Toeplitz matrix has eigenvalues (⌫ m ) m=0,...,p such that ⌫ 1 ⌫ m for all m =2,...,p with n =2p or n =2p +1.

Proof. Since the shape of the eigenvalues changes with the parity of n, let's again distinguish the cases.

For n odd, ⌫ 1 ⌫ m is equivalent to showing La quatrième contribution présente le problème de sériation avec duplication, motivé par l'assemblage de génomes cancéreux via des données de conformation spatiale, que nous tentons de résoudre avec un algorithme de projections alternées fondé en partie sur les méthodes de sériation robuste, sur données synthétiques.

Mots-clés

sériation, méthodes spectrales, optimisation combinatoire, relaxations convexes, permutations, permutaèdre, optimisation robuste, assemblage de novo , séquençage de troisième génération, Oxford Nanopore Technology, Overlap-Layout-Consensus, classement.

Abstract

In a sequencing experiment, we can only "read" small fragments (reads) of DNA due to physical limitations, whose location on the genome is unknown. De novo assembly aims to put them together to retrieve the full DNA sequence, like a jigsaw puzzle. The OLC approach computes pairwise Overlaps between reads to find their Layout, and then derive a Consensus sequence.

The layout can be cast as an instance of the Seriation combinatorial problem, seeking to reorder a set of elements based on their pairwise similarity, such that similar elements are nearby. In a noiseless setting, a spectral method can solve Seriation efficiently. Still, it often fails on noisy, real DNA data. Notably, assembly is challenged by repeated genomic regions (repeats) causing distant fragments to be similar. Most assembly engines follow hierarchical, greedy schemes, including modules dedicated to detect and disambiguate repeats while constructing the output sequence. We explore a simpler approach using Seriation to lay out all reads at once. Our first contribution is to show that the spectral method can be seamlessly integrated in an OLC framework, yielding competitive results compared to standard methods on real data. However, due to repeats, the method can only find fragmented assemblies (with a few large assembled fragments), i.e., it does not succeed to layout all the reads together at once. In our second contribution, we extend the spectral method using a multidimensional spectral embedding. It provides a unifying framework for seriation and circular seriation, a variant searching for a cyclic ordering of the data. This method significantly improves the robustness of the original algorithm on noisy data, and yields singlecontig assembly of bacterial genomes. As a third contribution, we introduce the Robust Seriation framework, formalizing the task of seriation on corrupted data. We outline the relation between (robust) seriation and other combinatorial problems, particularly for stylized matrices modeling DNA sequencing data. We propose dedicated algorithms that experimentally improve robustness on synthetic and real data, although they turn out to be more sensitive than the method constituting our second contribution. In a fourth contribution, we introduce the problem of Seriation with Duplications, which is motivated by the application of assembling cancer genome from spatial conformation (Hi-C) data. We propose an alternated minimization algorithm that can utilize methods designed to solve Robust Seriation, and evaluate it on toy data.

Keywords

seriation, spectral methods, combinatorial optimization, convex relaxations, permutations, permutahedron, robust optimization, de novo genome assembly, third generation sequencing, Oxford Nanopore Technology, overlap-layout-consensus, layout problems, ordering.

Figure 1 . 1 :

 11 Figure1.1: Similarity matrix between teapots images using the `2 distance, when the subscripts follow the ordering of the movie (1.1a), and when it is randomly permuted (1.1b).).

Figure 1 .

 1 Figure 1.2 shows a sub-sample of the ordered set of images found by seriation, on which we can see the rotative movement.

Figure 1 . 2 :

 12 Figure 1.2: Five teapots images sampled along the ordering found by seriation. On the top Figure 1.2a, the images are uniformly sampled along the ordering (at positions 1, 21, 41, 61, 81 out of 100). The bottom Figure 1.2b displays five consecutive images in the ordering. The mdso software presented in Chapter 3 was used in this experiment.

  Figure 1.3 illustrates these definitions.

Figure 1 . 3 :

 13 Figure 1.3: An R-matrix, A (3.2a) and a permuted observation, ΠAΠ T , with Π a permutation matrix (1.3b). Seriation seeks to recover the R-matrix (3.2a) from the pre-R matrix (1.3b).

  (1.1) are equal, then the right-hand-side is0.T o summarize, let 0= 1  2  ...  n , Λ , diag ( 1 ,..., n ), Φ =( 1 = f 1 ,...,f n ),b et h e eigendecomposition of L = ΦΛΦ T .The difficulty in solving 2-SUM (quad.) does not come from the objective function (the unconstrained minimization of a quadratic is one of the easiest optimization problems), but from the constraint set (combinatorial). As observed by Lim and Wright [2014], a permutation vector can be characterized by the three following constraints,⇡(i) 2 [n],i =1,...,n (integer constraint) ⇡ T 1 = n(n + 1)/2 (sum constraint)k⇡k 2 2 = n(n + 1)(2n + 1)/6. (norm constraint)

Theorem 1 . 2 . 1 Theorem 1 . 2 . 1 .

 121121 Atkins et al. [1998,T h e o r e m3 . 3 ]L e tA be a pre-R matrix with a simple Fiedler value and whose Fiedler vector f ⇤ has no repeated values. Let ⇡ be the permutation obtained by sorting the values of f ⇤ by increasing value (the output of Algorithm 1.1), and Π the associated permutation matrix. Then, ΠAΠ T is an R-matrix.

  are 2 n subsets S of [n], hence it is impractical to enforce explicitly all 2 n inequalities from (partial sum constraints) in a convex optimization solver. Lim and Wright [2014] proposed a convex relaxation of 2-SUM in PH, using an extended formulation of PH n from Goemans [2014], based on sorting networks. This formulation represents PH n with Θ(n log n) variables and constraints, instead of Θ(n 2 ) for permutation matrices.

Finally,

  Evangelopoulos et al. [2017a] also use the above regularization, i.e., they minimize the objective from (Vector-Relax. 2-SUM)i nPH n with the Frank-Wolfe Algorithm 1.2.

DNA sequencing refers toFigure 1 . 4 :

 14 Figure 1.4: Illustration of the genome assembly process. The long, black sequence is the DNA strand we wish to sequence. The colored fragments (called reads ) are the input for the assembly. However, these fragments are given "in a bag", i.e., we ignore their positions and their order in the genome (they are correctly located in the Figure for illustration only). Thanks to the overlaps, we can recover the full sequence from the fragments.

Figure 1 . 5 :

 15 Figure 1.5: Double-helix 3D structure of DNA. The DNA has two strands, where each C base is bounded with G,a n dT with A. One strand is the reverse component of the other, i.e.,i t is obtained by reading it backwards and replacing all Csb yGsa n da l lT sb yAs (and viceversa). In a genome assembly experiment, the reads may come from either strand. Therefore, one has to consider both orientations (strands) for each read when determining the layout. It adds consistency constraints on the layout, and is necessary to perform consensus. Figure from BCcampus on https://opentextbc.ca

Figure 1 . 6 :

 16 Figure1.6: Genome rearrangement around a repeat that occurs in three places. The reads at the junction between one of the two sequences A, B and a repeat R have apparent overlaps with the two other repeats, resulting in two possible layouts consistent with the overlaps constraints (the correct and the mis-assembled).

Figure 1 .

 1 Figure 1.6, a read that joins the end of a repeat and of another region in the genome will have apparent overlaps with the other repeats. For instance, on Figure 1.6, the read a 1 overlaps with r 1 ,b u ta l s or 3 .A n a l o g o u s l y ,b 1 overlaps with r 3 ,b u ta l s or 1 . In the end, the assembled layout [bottom] is consistent with the overlaps, yet it is incorrect (compared to the original sequence [top]).From an algorithmic perspective, the repeats may compromise the application of Seriation to genome assembly. Indeed, they induce overlaps between reads that can be far apart in the genome. For instance, in Figure1.6, b 2 overlaps with r 1 , but not with b 1 , a 2 nor a 1 . Yet, b 2 is further away from r 1 than from a 2 and a 1 . The resulting, correctly ordered similarity matrix therefore violates the Robinson property from Definition 1.1.1.

Figure 1 .

 1 Figure 1.7a shows the overlap-based similarity matrix between reads from a bacterium (Escherichia coli ). There are almost n ' 20000 reads, with a coverage c ' 30X. Therefore, with a repeat-free genome we would expect a similarity matrix Robinsonian and roughly banded with a bandwidth of order 30. Still, we observe a few out-of-band terms on Figure 1.7a,d u e to repeats. The resulting ordering found by the spectral Algorithm 1.1 shown in Figure 1.7b is corrupted. A complete introduction to shotgun sequence assembly, presenting the repeats problem, assembly paradigms and engineering challenges in details can be found in Pop [2004], Nagarajan and Pop [2013].

Figure 1 . 7 :

 17 Figure 1.7: Similarity matrix between reads from an E. coli genome (1.7a), and the ordering found with Algorithm 1.1 vs the true ordering (1.7b). If the ordering found was identical to the reference ordering, we would observe a straight line. The mis-assembly is imputable to the out-of-diagonal points observed in 1.7a.

1. 3 . 5

 35 Hi-C: Spatial Conformation Data Besides standard DNA sequencing techniques, an interesting recent development called Hi-C and based on the chromosome conformation capture (3C) technology allows to measure experimentally the frequency of physical interactions in 3D between all pairs of positions in the genome [Lieberman-Aiden et al., 2009a].

Figure 2 . 1 :

 21 Figure 2.1: A similarity matrix reordered with the spectral algorithm. The original matrix (left) has values that decrease when moving away from the diagonal. It is randomly permuted (right), and the spectral algorithm will find back the original ordering.

Figure 2 . 2 :

 22 Figure 2.2: Consensus generation. Given the layout, the genome is sliced into overlapping windows, and a consensus is computed in each window. The final consensus is then obtained by merging the consensus windows.

Figure 2 . 3 :

 23 Figure 2.3: Similarity matrix for E. coli ONT sequences before (left) and after (right) thresholding. The positions of the reads were obtained by mapping to the reference genome with GraphMap [Sović et al., 2016].

Algorithm 2 . 2 1 : 2 : 3 : 3 ) 4 :

 2212334 OLC assembly pipelineInput: n long noisy reads Compute overlaps with an overlapper (e.g. minimap or MHAP) Construct similarity matrix S 2 R n⇥n from the overlaps Remove outliers from S with a threshold on values S ij , on overlap length, and removal of connecting reads (as explained in §2.2.for all Connected component A of S do 5:

Figure 2 . 4 :

 24 Figures A.6 and A.7 and Tables A.2 and A.4.

Figure 2 . 6 :

 26 Figure 2.6: Ordering of the Saccharomyces cerevisiae ONT R7.3 reads identified with the spectral algorithm vs true ordering (obtained by mapping the reads to the reference genome with GraphMap and concatenating the ordering found in each chromosome). The different chromosomes are separated by grid lines.

and A. 5

 5 highlight the correcting effect of the consensus. Figure2.7 provides hints about what causes inaccurate consensus windows, and suggests that the error-rate in the consensus windows depends mainly on the local coverage. The top plots examine the error-rate in the consensus windows according to their position (and whether they are located on a repeat). Most of the windows with a high error rate are positioned at the ends of the contigs to which they belong.

3 Figure 2 . 7 :

 327 Figure 2.7: Error-rates in consensus windows versus position of the windows on the reference genome (a,b). The dashed lines represent the location of repeats for A. baylyi, and the separation between chromosomes for S. cerevisiae. The size of each scatter marker is proportional to the coverage of the window. The(c,d) panel represents the error-rates in consensus windows versus the coverage of the windows. The error-rate was computed with the errorrates.py script from samtools, using the mapping obtained from GraphMap.

Figure 2 . 8 :

 28 Figure 2.8: Error rate of consensus window sequences, compared to the raw and corrected (with the Canu correction and trimming modules) reads for the A. baylyi ONT dataset. The error rates were computed by mapping the sequences to the A. baylyi reference genome. Histograms for the other datasets are available in Supplementary Figure A.5.

( 3 Figure 2 . 9 :

 329 Figure 2.9: Histograms of the number of contigs as a function of the number of distinct restriction sites (RS) appearing in their sequence (a) or contig length (b). For a given number of RS occurrences (a) or contig length (b), the blue part of the bar shows the fraction of contigs correctly aligned to the theoretical restriction map, whereas the red part corresponds to the complementary fraction of unperfectly aligned contigs.

Figure 3 . 1 :

 31 Figure3.1: Illustration of the assembly process for a circular genome. The physical strand of DNA is circular. The DNA sequence is therefore represented as a circle (black). Reads are randomly sampled from this sequence, and we wish to infer their position (layout) from their pairwise overlaps. Given an ordering of the reads matching their position on the (linearized) genome, the first and the last reads are likely to overlap. Here, the first read (dark blue) overlaps with the last read (red).

  f o ra llt r i p l e t so fi n d i c e s( i , j , k )s u c ht h a tD(|i j|) D(|i k|),w h e r eD(|i j|)= min(|i j|,n |i j|). As a reminder, S n is the set of real symmetric matrices of dimension n. The proximity matrix of points embedded on a line follows Definition 3.1.1, whereas that of points embedded on a circle (as in Figure B.1) follows Def 3.1.2.F i g u r e3.2 displays examples of such matrices.

Figure 3 . 2 :

 32 Figure 3.2: From left to right, R-matrix (3.2a), circular R-matrix (3.2b), and a randomly permuted observation of a R-matrix (3.2c). Seriation seeks to recover (3.2a) from its permuted observation (3.2c).

Algorithm 3 .

 3 1 embeds the data in 1D through the eigenvector f 1 (1-LE). It then uses this 1-d embedding to infer an ordering of the points. More generally, for any d<n , Φ (d) , (f 1 ,...,f d ) defines a d-dimensional embedding (d-LE)

  The full (↵,d ) -LE, with ↵ k =( rw k ) 1/2 and d = n 1, satisfies CTD(i, j) /k ỹi ỹj k. Given the decay of ↵ k ,t h ed -LE with d ⌧ n approximately preserves the CTD. This embedding has been successfully applied to vision tasks, e.g., anomaly detection [Albano and Messinger, 2012], image segmentation and motion tracking [Qiu and Hancock, 2007]. Another, closely related dimensionality reduction technique is that of diffusion maps [Coifman and Lafon, 2006], where the embedding is derived to preserve diffusion distances, resulting in the (↵,d ) -LE, for t 0, ↵ k (t)=(1 rw k ) t . Coifman and Lafon [2006], Coifman et al. [2008] also propose a normalization of the similarity matrix W D 1 WD 1, to extend the convergence of L rw towards the Laplace-Beltrami operator on a curve when the similarity is obtained through a heat kernel on points that are non uniformly sampled along that curve.Finally, we will use in practice the heuristic scaling ↵ k =1 / p k to damp high dimensions, as explained in Appendix B.2.5.For a deeper discussion about spectral graph theory and the relations between these methods, see for instanceQiu and Hancock [2007]a n dChung and Yau[2000].

  the graph Laplacian L of the heat kernel (W ij = exp kx i x j k 2 /t ) used in Belkin and Niyogi [2003] is a discrete approximation of ∆ M , the Laplace-Beltrami operator on M (a differential operator akin to the Laplace operator, adapted to the local geometry of M). Singer [2006] specify the hypothesis on the data and the rate of convergence of L towards ∆ M when n grows and the heat-kernel bandwidth t shrinks. Von Luxburg et al. [2005] also explore the spectral asymptotics of the spectrum of L to prove consistency of spectral clustering. This connection with continuous operators gives hints about the Laplacian embedding in some settings of interest for Linear Seriation and Circular Seriation. Indeed, consider n points distributed along a curve Γ ⇢ R D of length 1, parameterized by a smooth function : R ! R D ,

[

  2008]u s et h e2 -LE to perform Circular Seriation in a tomographic reconstruction setting, sorting the inverse tangent of the angle between the two components to reorder the points (Algorithm 3.2). Liu et al. [2018]u s eas i m i l a ra p p r o a c ht os o l v eCircular Seriation in a

  Let us consider the set C ⇤ R of matrices in C R that are circulant, in order to have a closed form expression of their spectrum. A matrix A 2 R n⇥n is Toeplitz if its entries are constant on a given diagonal, A ij = b (i j) for a vector of values b of size 2n 1. A symmetric Toeplitz matrix A satisfies A ij = b |i j| , with b of size n. In the case of circulant symmetric matrices, we also have that b k = b n k , for 1  k  n, thus symmetric circulant matrices are of the form,

1 : 1 A 2 : 3 :

 1123 through m times, hence the ordering of the points on the circle does not follow their latent ordering. The ⌫ m from equations (3.4) are in general not sorted. It is the Robinson property (monotonicity of (b k )) that guarantees that ⌫ 1 ⌫ m , for m 1,andthusthatthe2-LE embeds the points on a circle that follows the latent ordering and allows one to recover it by scanning through the unit circle. This is formalized in Theorem 3.3.2, which is the main result of our paper, proved in Appendix B.3. It provides guarantees in the same form as in Theorem 3.2.1 with the simple Algorithm 3.2 that sorts the angles, used in Coifman et al. [2008]. Algorithm 3.2 Circular Spectral Ordering [Coifman et al., 2008] Input: Connected similarity matrix A 2 R n⇥n Compute normalized Laplacian L rw A = I diag(A1) Compute the two first non-trivial eigenvectors of L rw A , (f 1 ,f 2 ) Sort the values of ✓

6 )

 6 Toeplitz matrices are defined by b 0 >b 1 > 0=b 2 = ... = b p .F o r m = 0,...,n 1, they have eigenvalues ⌫ m with multiplicity 1 associated to eigenvector y (m) [Trench, 1985], ⌫ m = b 0 +2b 1 cos m⇡/(n + 1) y (m) = ⇣ sin m⇡/(n + 1) ,...,sin mn⇡/(n + 1) spectrum of the Laplace operator on a curve with endpoints from §3.2.3 (up to a shift). This type of matrices can indeed be viewed as a limit case with points uniformly sampled on a line with strong similarity decay, leaving only the two nearest neighbors with non-zero similarity. Kac-Murdock-Szegö (KMS) matrices are defined, for ↵ > 0, ⇢ = e ↵ ,b yA ij = b |i j| = e ↵|i j| = ⇢ |i j| .F o rm =1 ,...,bn/2c, there exists ✓ m 2 (m 1)⇡/n, m⇡/n ,s u c ht h a t⌫ m is a double eigenvalue associated to eigenvectors y m,cos ,y m,sin , Linearly decreasing Toeplitz matrices defined by A lin ij = b |i j| = n |i j| have spectral properties analog to those of KMS matrices (trigonometric expression, interlacement, low frequency assigned to largest eigenvalue), but with more technical details available in Bünger [2014]. This goes beyond the asymptotic case modeled by tridiagonal matrices. Banded Robinson Toeplitz matrices typically include similarity matrices from DNA sequencing. Actually, any Robinson Toeplitz matrix becomes banded under a thresholding operation. Also, fast decaying Robinson matrices such as KMS matrices are almost banded. There is a rich literature dedicated to the spectrum of generic banded Toeplitz matrices [BoeÓttcher and Grudsky, 2005, Gray et al., 2006, Böttcher et al., 2017]. However, it mostly provides asymptotic results on the spectra. Notably, some results indicate that the eigenvectors of some banded symmetric Toeplitz matrices become, up to a rotation, close to the sinusoidal, almost equi-spaced eigenvectors observed in equations (3.5)a n d( 3.6)[ Böttcher et al., 2010, Ekström et al., 2017].

line 7

 7 we can update the similarity S uv by adding any non-increasing function of the distance D uv , e.g., D 1 uv , exp ( D uv ),o r D uv (the latter case requires to add an offset to S afterwards to ensure it has non-negative entries, and is what we implemented in practice.) In line 9,t h e matrix S needs to be connected in order to use Algorithm 3.1, which is not always verified in practice (for low values of k, for instance). In that case, we reorder separately each connected component of S with Algorithm 3.1, and then merge the partial orderings into a global ordering by using the input matrix A, as detailed in Algorithm B.1, Appendix B.1.

3

 3

Figure 3 . 3 :

 33 Figure 3.3: Noisy Circular Banded matrix (3.3a) and associated 2d Laplacian embedding (3.3b).

Figure 3 . 4 :

 34 Figure 3.4: Matrix S created through Algorithm 3.3 (3.4a), and associated 2d-Laplacian embedding (3.4b).

2 -

 2 LE under a perturbation of the matrix A using the Davis-Kahan theorem [Davis and Kahan, 1970], well introduced in [Von Luxburg, 2007, Theorem 7], yielding the following result.

3. 5 . 1

 51 Application of the Davis-Kahan Theorem Proposition 3.5.1 (Davis-Kahan). Let L and L = L + L be the Laplacian matrices of A 2 C ⇤ R and A + A 2 S n ,r e s p e c t i v e l y ,a n dV, Ṽ 2 R 2⇥n be the associated 2-LE of L and L,i . e . ,

Proposition 3 . 5 . 2 (

 352 Davis-Kahan). Consider L ag r a p hL a p l a c i a no faR -s y m m e t r i c -c i r c u l a r Toeplitz matrix A.W ea d das y m m e t r i cp e r t u r b a t i o nm a t r i xH and denote by à = A + H and L the new similarity matrix and graph Laplacian respectively. Denote by (p i ) i=1,...,n and (p i ) i=1,...,n the 2-LE coming from L and L respectively. Then there exists a cyclic permutation ⌧ of {1,...,n} such that sup i=1,...,n

Figure 3 . 5 :

 35 Figure 3.5: Kendall-Tau scores for Linear (3.5a) and Circular (3.5b) Seriation for noisy observations of banded, Toeplitz, matrices, displayed for several values of the dimension parameter of the d-LE(d), for fixed number of neighbors k = 15.Finally, in Figure3.6, we compare our Algorithm 3.3 when using alternative embedding methods mentioned in §3.2.4, namely classical MDS (denoted cMDS), metric-MDS (denoted MDS), and t-SNE, instead of the spectral (Laplacian embedding). The method performs similarly when used with a classical-MDS or spectral embedding, which is not surprising since both method rely on the spectral decomposition of slightly differently normalized similarity matrices. The spectral embedding outperforms the other techniques in these experiments, empirically justifying the choice of embedding made from theoretical considerations.

Figure 3 . 6 :

 36 Figure 3.6: Kendall-Tau scores for Seriation for noisy observations of Linear and Circular banded, Toeplitz, matrices, displayed for several embedding methods. All embeddings are of dimension d =8except t-SNE for which d =2. The number of neighbors is set to k = 15.

  baylyi. The new similarity matrix S computed from the embedding in Algorithm 3.3 was disconnected, resulting in several connected component instead of one global ordering (see Figure B.2b).

Figure 3 . 7 :

 37 Figure 3.7: Overlap-based similarity matrix from E. coli (3.7a)a n dA. baylyi (3.7c) reads, and the ordering found with Algorithm 3.3 (for E. coli -3.7b,a n dA. baylyi -3.7d) versus the position of the reads within a reference genome obtained by mapping to a reference with minimap2. The genome being circular, the ordering is defined up to a shift, which is why we observe two lines instead of one in (3.7b and 3.7d).

[

  [START_REF] Dekker | Exploring the three-dimensional organization of genomes: interpreting chromatin interaction data[END_REF].Interestingly, the frequency of interactions between genomic loci (called bins) tend to decrease with their distance in the linear genome[Lieberman-Aiden et al., 2009b]. Also, interactions are more frequent within a given chromosome than between distinct chromosomes. Thus, we can use Hi-C data to find the layout of the bins in a de novo assembly experiment[Dudchenko et al., 2017]. We have seen that due to repeats, reads-overlap-based similarity matrices had a specific structure (banded Robinsonian matrices + sparse out-of-band noise) challenging seriation methods. Hi-C similarity matrices are also expected to be close to Robinsonian matrices, but with a different structure. The underlying stylized Robinsonian matrices are not banded but harbour a power-law decay when moving away from the diagonal, and the noise seem to have another structure that repeat-induced noise.

  From equations (3.1)a n d (Relax. 2-SUM), we can see that if the input similarity matrix is disconnected into K connected components, then the eigenvalue 0 has multiplity K +1, with associated eigenvectors 1 and the indicator vectors C k of the K connected components C k ,k=1 ,...,K. Indeed, consider a given connected component C k . By definition, A ij =0if i 2 C k and j/ 2 C k . Thus, if f = C k in the objective from equation (3.1), then for each pair (i, j), either i, j 2 C k and

  spectral, d =1, 2 (c) spectral, d = 11, 12

Figure 3 . 8 :

 38 Figure 3.8: t-SNE 2D embedding(3.8a), and two projections of the spectral embedding (3.8b, 3.8c) for the synthetic multiple chromosomes frequency matrix DL1. The colormap goes from dark blue to yellow with the absolute position of the bins, where the first bin is the first bin in the first chromosome, and the last bin is the last bin from the last chromosome (with an arbitrary ordering between chromosomes for illustrative purposes).

Figure 3 . 9 :

 39 Figure 3.9: sub-orderings found by mdso on synthetic, multiple chromosomes Hi-C data.

  10 and Appendix Figure B.9, where we show the tSNE embedding and the resulting sub-orderings in two instances of the experiment, one leading to ⌧ = 91.8,a n do n et o⌧ = 61.6, where the low

Figure 3 .

 3 Figure 3.10: t-SNE embedding (3.10a), and resulting sub-orderings found with mdso (3.10b) on the Plasmodium knowlesi Hi-C data, in an experiment leading to a weighted Kendall-Tau score of 91.8%. The line ticks in (3.10b) delimitate the chromosomes.

  Liu et al. [2017] used it in order to cluster each cells according to four possible cell-cycle phases (G1, E-S, M-S or L-S/G2). Specifically, they applied classical MDS to HiCRep[Yang et al., 2017b] data, approximately embedding it onto a circle. They introduced a circular-ROC (CROC) measure to assess the ability of the embedding to distinguish between the four phases. This circle-like embedding reflects a latent ordering on the data, each capturing a cell architecture at a given stage of the cell life. Although we are ultimately interested in a clustering task, it can benefit from an embedding enhancing the latent ordering of the data, as Algorithm 3.3 produces.

Figure 3 .

 3 Figure 3.11: Various embedding methods of HicRep data. Figure 3.11a is the first two dimension of Multi-dimensional Embedding while Figures 3.11b and 3.11c are the first two dimensions of the embedding resulting from Algorithm 3.3 for k = 10 and k = 20 respectively.

  Definition 4.1.1 is more restrictive than the usual R-matrix property from Definition 1.1.1 (repeated in Definition 3.1.1), and used in Atkins et al. [1998], Fogel et al. [2013], which only requires the entries of the matrix to decrease when moving away from the diagonal on a given row or column.F o r s t r o n g -R matrices, we impose that the entries on a given diagonal are no greater than any entry located on the previous diagonals (see Figure 4.1).

Figure 4 . 1 :

 41 Figure 4.1: A R-matrix (4.1a) and its projection on the set of strong-R matrices (4.1b). A pre-strong-R matrix (4.1c) is a strong-R matrix up to a permutation of the rows and columns. If a matrix is pre-strong-R (4.1c), Seriation aims to find the permutation that makes it strong-R (4.1b).

  ].More recently, several relaxations have been proposed to tackle 2-SUM and QAP(A,B),a lthough there are no approximation bounds in the general case[Lyzinski et al., 2016]. Vogelstein et al. [2011] used the Frank-Wolfe algorithm to minimize the objective of QAP(A,B) over the convex hull of the permutation matrices, namely the Birkhoff polytope B. Fogel et al. [2013]presented a convex relaxation of 2-SUM in B, and used a quadratic programming approach where the variable's membership to B is enforced through linear constraints (instead of the implicit projection of the Frank-Wolfe algorithm). Lim and Wright [2014] proposed a similar relaxation in the convex hull of the set of permutation vectors, the Permutahedron PH n , represented with Θ(n log n) variables and constraints, instead of Θ(n 2 ) for permutation matrices, thanks to an extended formulation by Goemans[2014]. All these relaxations for 2-SUM suffer from a symmetry problem, because flipping permutations leaves the objective unchanged, and the minimum of 2-SUM is achieved for a vector proportional to 1 =( 1 ,...,1) T , which lies in the center of the convex hull of permutation vectors. To overcome this issue, constraints can be added to the problem, corresponding to either apriorikwowledge, or to pure "tie-breaking", e.g.,⇡ 1 +1 ⇡ n ,ensuring that the center is excluded from the constraint set, thus breaking symmetry without loss of generality. Lim and Wright [2014] stated that a Frank-Wolfe algorithm could also be used for 2-SUM in PH if no other constraint but the tie-breaking was enforced, thanks to a specific linear minimization oracle, thus implicitly enforcing membership to PH without imposing the constraints from Goemans [2014]. Lim and Wright [2016] generalized the use of the representation of Goemans [2014] for PH to tackle QAP(A,B), with a coordinate descent algorithm and a continuation scheme to move away from the center of the convex hull of permutations. Evangelopoulos et al. [2017a] proposed a Frank-Wolfe algorithm in PH with a continuation scheme (instead of a tie-breaking constraint) to tackle 2-SUM and avoid the center. They also discussed problems of the form (QAP(A,B)) where B ij = Pseudo-Huber(|i j|) [Evangelopoulos et al., 2017b], which helps in solving robust seriation as we will see below.

Figure 4 . 6 from Chapter 1 .

 461 Figure 4.2a, which is a subset of the matrix shown in Figure 1.7a.Repeats longer than the overlap length are perhaps the most fundamental issue in genome assembly as they lead to ambiguous reconstructions. For instance, recall the sequence RARBR, where A,B and R are sub-sequences, and R is repeated three times, illustrated in Figure1.6 from Chapter 1. The overlap constraints arising from this sequence are identical to those of RBRAR, therefore the overlap constraints are not sufficient to uniquely determine the layout.

Figure 4 . 2 :

 42 Figure 4.2: Similarity matrix from a subset of Oxford Nanopore reads of E. coli in the ordering given by the ground truth position of the reads along the genome (4.2a, left), and the same matrix reordered by minimizing the 2SUM objective (4.2b, right), which pushes the out-ofdiagonals terms close to the main diagonal and yields a corrupted ordering.

observe that b 2 R

 2 n+1 has integer entries, and that A 2 R (n+1)⇥(n+1) is totally unimodular. It follows that it has an integral solution x ⇤ [Papadimitriou and Steiglitz, 1998][Th. 13.3].

Figure 4 . 3 :

 43 Figure 4.3: Plot of the square (`2), absolute value (`1), Huber, and truncated square losses, appearing respectively in the (2-SUM), (1-SUM), (HuberSUM( )), and (R2SUM( )) problems.

  Vogelstein et al. [2011]a n dLim and Wright [2016] methods for solving the QAP. The first, which we refer to as FAQ [Vogelstein et al., 2011], uses the matrix representation of permutations with a relaxation in the convex hull of permutation matrices, B, where the QAP(A,B) objective is optimized with the conditional gradient (a.k.a. Frank-Wolfe) algorithm, described in the Introduction Chapter, Algorithm 1.2, and repeated here in 4.1. Each step of Frank-Wolfe in B (4.2) involves an assignment problem solved with a Hungarian algorithm [Kuhn, 1955].

Algorithm

Figure 4 . 4 :

 44 Figure 4.4: View of the 3-Permutahedron PH 3 (filled polygon) in the 2D plane H 3 (orthogonal to the vector 1 3 represented by the red pointing arrow (circled dot)). The blue ellipses are the level curves of f 2SUM . The black dashed circle represents the set of points having the same norm as the permutation vectors, and the black diamond is the minimizer of 2-SUM among them. The green (resp. orange) line is where the "good" (resp. "bad") tie-breaking constraint ⇡ 2 +1  ⇡ 3 (resp. ⇡ 1 +1  ⇡ 3 ) is active, and the green (resp. orange) diamond is the minimizer of f 2SUM on the corresponding constrained set, the triangle (2, 1, 3),(1, 2, 3),(1, 3, 2) [resp.(3, 1, 2), (2, 1, 3), (1, 2, 3)]. The closest permutation to the green diamond is (2, 1, 3), which is the correct solution (minimizer of f 2SUM on P 3 ), but the orange diamond is closer to (1, 2, 3) because of the anisotropy induced by the tie-breaking constraint. Figure adapted from Lim and Wright [2014].

  Figure 4.4: View of the 3-Permutahedron PH 3 (filled polygon) in the 2D plane H 3 (orthogonal to the vector 1 3 represented by the red pointing arrow (circled dot)). The blue ellipses are the level curves of f 2SUM . The black dashed circle represents the set of points having the same norm as the permutation vectors, and the black diamond is the minimizer of 2-SUM among them. The green (resp. orange) line is where the "good" (resp. "bad") tie-breaking constraint ⇡ 2 +1  ⇡ 3 (resp. ⇡ 1 +1  ⇡ 3 ) is active, and the green (resp. orange) diamond is the minimizer of f 2SUM on the corresponding constrained set, the triangle (2, 1, 3),(1, 2, 3),(1, 3, 2) [resp.(3, 1, 2), (2, 1, 3), (1, 2, 3)]. The closest permutation to the green diamond is (2, 1, 3), which is the correct solution (minimizer of f 2SUM on P 3 ), but the orange diamond is closer to (1, 2, 3) because of the anisotropy induced by the tie-breaking constraint. Figure adapted from Lim and Wright [2014].

⇡ 2 Proposition 4 . 3 . 1 .

 2431 [2014]. Yet, while Fogel et al. [2013], Lim and Wright [2014] proposed convex optimization methods that could incorporate any such linear constraint into the problem seamlessly, if one wants to use Frank-Wolfe in the restriction of PH where the tie-break is satisfied, the LMO has to be modified. The new LMO must solve, PH n , ⇡ i +1 ⇡ j . (LMO-tb) where we let 1  i 6 = j  n be the tie-break indexes (in Fogel et al. [2013], Lim and Wright [2014], i =1and j = n). Lim and Wright [2014] propose an algorithm for solving LMOthat preserves the O(n log n) complexity of the LMO. We describe in Algorithm 4.2 as l i g h t l y simplified version of theirs, for any tie-break indexes 1  i 6 = j  n.W eu s et h em a t l a b -l i k e notation x(i) to denote x i for ease of reading. Algorithm 4.2 minimizes g T ⇡ over PH n with tie-break ⇡(i)+1 ⇡(j).

7 :

 7 Set z =(1,...,k 1,k+2,...,n) T 2 R n 2 8:

  Non-Convexity : Frank-Wolfe Algorithm with Concave Penalty (GnCR and HGnCR) In Fogel et al. [2013], Lim and Wright [2014], the parameter µ controlling the amplitude of the penalty µ kx ck 2 is bounded in order to keep the objective convex. Precisely, the objective

Figure 4 . 5 :

 45 Figure 4.5: Illustration of Algorithm 4.3 in the 3-Permutahedron PH 3 (filled polygon, same representation as in Figure 4.4). The colored crosses (from flashy yellow (right) to red (left)) represent the solutions x (t)

Figure 4 . 6 :

 46 Figure 4.6: Ordering found with the spectral baseline Algorithm 4.4 (4.6a), and with the ⌘-Spectral Algorithm 4.5 (4.6b)o nt h eE. coli ONT data.

Figure 4 . 7 :

 47 Figure 4.7: Ordering found with the spectral baseline Algorithm 4.4 (4.7a), and with the ⌘-Spectral Algorithm 4.5 (4.7b)o nt h eA. baylyi ONT data..

  Reconstructing the 1D structure of a cancer genome from experimental data is an important problem. Besides standard DNA sequencing techniques, an interesting recent development called Hi-C and based on the chromosome conformation capture (3C) technology allows to measure experimentally the frequency of physical interactions in 3D between all pairs of positions in the genome [Lieberman-Aiden et al., 2009a]. In short, if we split the full human genome into n bins (of typical length 10 4 10 6 basepairs each), an Hi-C experiment produces an n ⇥ n

Figure 5 . 1 :

 51 Figure 5.1: Structure of a typical cancer genome (breast cancer cell line). Instead of the standard 23 pairs of chromosomes, cancer cells often harbour large structural variants, such as changes in copy number and translocations. Reconstructing this 1D map from high-throughput Hi-C or sequencing data is an important problem that motivates the definition of seriation with duplications. Figure from Karp et al. [2015].

  frequency of interactions between fragments k and l, respectively extracted from bins b i and b j , what Hi-C measures as interactions between b i and b j is the sum of S k 0 l 0 where k 0 and l 0 are fragments contained in b i and b j , respectively. This motivates the definition of the seriation with duplication problem formalized below.

  as in clustering problems). Such an assignment matrix is linked to the vector-based notation ⇡ 2 [1,n] N from above through ⇡ = Z T (1, 2,...,n) T . We write Z c the set of assignment matrices for a given duplication count vector c 2 N n ,Z c = n Z 2 {0, 1} n⇥N Z1 N = c, Z T 1 n = 1 N o where N = c T 1 n ,and the constraints indicate that each bin i 2 [1,n] has c i duplicates, and that each element k 2 [1,N] comes from one single bin. Observe that given an initial assignment matrix Z 0 2 Z c , any other Z 2 Z c can be expressed as Z 0 whose columns have been permuted, i.e. there exists Π 2 P N such that Z = Z 0 Π.A si nt h eSeriation formulation, the problem of Seriation with Duplications can be written find Π 2 P N ,S2 R ⇤ N such that Z 0 ΠSΠ T Z T 0 = A.(SD)

3 . 3 . 3 . 5 . 1

 33351 Projecting onto the set of matrices satisfying linear equality constraints in step 4 can also be done with a convex programming solver, but the problem is actually separable on the values (i, j) 2 [1,n] ⇥ [1,n] and has a closed form solution detailed in Section 5.3.Algorithm General Alternating Projection Scheme for Seriation with Duplications.

  power law of the distance to the diagonal, S kl = |k l| , which is consistent with the observed frequency of intra-chromosomal interactions [Lieberman-Aiden et al., 2009b]. We used N = 200 and tried several values for the exponent and the ratio N/n, namely 2 {0.1, 0.5, 1} and N/n 2 {1.33, 2, 4}. The results are shown in Tables 5.2, D.2, D.3,a n ds o m eq u a l i t a t i v e results are shown in Figure D.1. We also conducted experiments with sparse, band matrices S 2 M N ( ,s) as in Section 4.2. The results are shown in Tables 5.1, D.4, D.5,a n ds o m e qualitative results are shown in Figure D.2.

Figure 5 . 2 :Algorithm 5 . 3

 5253 Figure 5.2: Similarity matrix with Robinson + Block structure (5.2a), and the associated cluster-crossing curve that enables one to determine the breakpoints between the clusters (5.2b).

  et al.[1998], which essentially rely on results specific to the ordering relation of real numbers.Moreover, in the linear seriation case, we provided insight about the curve structure of the Laplacian embedding through the study of the spectrum of specific Toeplitz matrices, but we could not generalize to general Robinson matrices. An interesting work direction is to search for theoretical guarantees about circular seriation under milder assumptions on the input matrices. Another line of work concerns the normalization of the Laplacian (or of the similarity matrix itself). Indeed, we have explored the normalization proposed by Coifman et al.[2008], that roughly normalizes the similarity by the local density of points. It would be interesting to compare this method to normalizations of the similarity matrix able to make it closer to a Toeplitz matrix, such as Sinkhorn-Knopp normalization.

  3 from Chapter 3 creates a new similarity matrix from a spectral embedding of the data. When there is a clear cluster structure in the data, it translates to the embedding and the new similarity matrix can be disconnected into connected components corresponding to individual chromosomes.Still, in most cases, with real data involving separate chromosomes, dividing the reads into clusters can be crucial, yet it is not explicitly handled by our methods. An interesting line of future research would be to formulate the task of performing both clustering and ordering simultaneously as an optimization problem over permutations, and try to derive dedicated algorithms. A possible lead could be to follow the approach of Lim and Wright [2016]using the extended formulation of the Permutahedron[START_REF] Michel | Smallest compact formulation for the permutahedron[END_REF].

Figure A. 1

 1 Figure A.1: Runtime of the spectral ordering algorithm in connected components of different sizes (across all datasets), with two solvers for the eigenvalues computations (scipy.sparse.eigsh and the eigs function from Julia [Bezanson et al., 2017]).

Figure A. 4 :

 4 Figure A.4: Similarity matrices containing outliers, displayed with true ordering (obtained by mapping the reads to the reference genome with GraphMap) and generated with a subset of A. baylyi ONT NanoSim perfect reads A.4a, and the same matrix incorrectly reordered with the spectral algorithm A.4b.

Figure A. 5 :

 5 Figure A.5: Error-rates in consensus windows, raw reads and corrected reads for the six real datasets.

Figure A. 5

 5 Figure A.5 is the analog of Figure 2.8 in Chapter 2. It shows the error rate in the raw reads, in the reads corrected with Canu, and in the consensus windows. With ONT R7.3 data, the consensus produced by our pipeline appears more accurate than via the correction module of Canu, while the contrary is true for PacBio data.

Figure A. 6 :

 6 Figure A.6: Ordering of the reads computed with the spectral algorithm vs true ordering (obtained by mapping the reads to the reference genome with GraphMap) for the original (a-c) and corrected (d-f) bacterial datasets. All contigs are artificially displayed on the same plot for compactness.

  Total reordered sequence c final , which is a permutation if C =1or a set of reordered subsequences if the loop broke at line 5.

Figure B. 1 :

 1 Figure B.1: Illustration of the shift-invariance of permutations solution to a Circular Seriation problem.

C data experiments B. 3

 3 Proof of Theorem 3.3.2

  's define Z n = {cos(2⇡k/n) | k 2 N}\{ 1; 1}. Depending on the parity of n, we will write n =2p or n =2 p +1. Hence we always have p = ⌅ n 2

Lemma B. 3 . 1 .S

 31 For z (m) k = cos( 2⇡km n ), n =2p +1 and any m =1,...,p

5 )

 5 Finally for n even and m odd we have S

1 )

 1 (m 1)/2 cos (x), if m is odd It follows that, for any m, x = ⇡m/(2n), this formula simplifies the numerator appearing in equation (B.7) and yields the result in equation (B.2).

4 .⇣ 1 ,⇣ 1 , 1 X= b 0 ⇣ 1 ,

 411101 Consider A, a circular-R Toeplitz matrix of size n. For n =2p +1 ⌫ m , b 0 +2 For m =1,...,p each ⌫ m are eigenvalues of A with multiplicity 2 and associated eigenvectors y m,cos = 1 p n cos 2⇡m/n ,...,cos 2⇡m(n 1)/n ⌘ y m,sin = 1 p n sin 2⇡m/n ,...,sin 2⇡m(n 1)/n ⌘ . (B.14)For n =2p ⌫ m , b 0 +2 P p 1 k=1 b k cos ⇣ 2⇡km n ⌘ + b p cos (⇡m) , (B.15)where ⌫ 0 is still singular, with y (0) = 1 p n (1,...,1) . ⌫ p also is, with y (p) = 1 p n (+1, 1,...,+1, 1) , and there are p 1 double eigenvalues, for m =1,...,p 1,e a c ha s s o c i a t e dt ot h et w oe i g e nvectors given in equation (B.14).Proof. Let us compute the spectrum of a circular-R, symmetric, circulant Toeplitz matrix. From Gray et al.[2006], the eigenvalues are⌫ m = n k=0 b k ⇢ k m , (B.16)with ⇢ m =exp( 2i⇡m n ), and the corresponding eigenvectors are,y (m) = 1 p n ⇣1,e 2i⇡m/n ,...,e 2i⇡m(n 1)/n ⌘ , (B.17) for m =0,...,n 1.Case n is odd, with n =2 p +1. Using the symmetry assumption b k = b n k , and the fact that⇢ n k m = ⇢ n m ⇢ k m = ⇢ k m , it results in real eigenvalues, ⌫ m = b 0 + P p k=1 b k ⇢ k m + P n 1 k=p+1 b k ⇢ k m ⌫ n m = ⌫ m ,for m =1,...,n 1, resulting in p +1 real distinct eigenvalues. ⌫ 0 is singular, whereas for m =1 ,...,p, ⌫ m has multiplicity 2, with eigenvectors y m and y n m . This leads to the two following real eigenvectors, y m,cos =1/2(y m + y n m ) and y m,sin = 1/(2i)(y m y n m ) y m,cos = 1 p n cos 2⇡m/n ,...,cos 2⇡m(n 1)/n ⌘ y m,sin = 1 p n ⇣ 1, sin 2⇡m/n ,...,sin 2⇡m(n 1)/n ⌘ (B.19)Case n is even, with n =2p. A derivation similar to (B.18) yields,⌫ m = b 0 +2 P p 1 k=1 b k cos ⇣ 2⇡km n ⌘ + b p cos (⇡m) (B.20)⌫ 0 is still singular, with y (0) = 1 p n (1,...,1) , ⌫ p also is, with y (p) = 1 p n (+1, 1,...,+1, 1) , and there are p 1 double eigenvalues, for m =1 ,...,p 1, each associated to the two eigenvectors given in equation (B.14).

b

  k cos(2⇡km/n) . (B.21) It is true by combining proposition B.3.3 with lemma B.3.2. The same follows for n even and m odd.

  Figure B.2: 3d Laplacian embedding from overlap-based similarity matrix of E. coli (B.2a) and A. baylyi (B.2c) reads, and the orderings found in each connected component of the new similarity matrix created in Algorithm 3.3 (B.2b and B.2d) versus the position of the reads within a reference genome obtained by mapping tge reads to the reference with minimap2 (all plotted on the same plot for compactness). The orderings have no absolute direction, i.e., (1, 2,...,n) and (n, n 1,...,1) are equivalent, which is why the lines in Figures B.2b and B.2d can be either diagonal or anti-diagonal.

  Figure B.3: K-T scores for Linear (B.3a) and Circular (B.3b) Seriation for noisy observations of KMS, Toeplitz, matrices, displayed for several values of the dimension parameter of the d-LE.

Figure B. 4 :

 4 Figure B.4: K-T scores for Linear (B.4a) and Circular (B.4b) Seriation for noisy observations of banded, Toeplitz, matrices, displayed for several values of the number of nearest neighbors k, with a fixed value of the dimension of the d-LE, d = 10.

  Figure B.5: K-T scores for Linear (B.5a) and Circular (B.5b) Seriation for noisy observations of KMS, Toeplitz, matrices, displayed for several values of the number of nearest neighbors k, with a fixed value of the dimension of the d-LE, d = 10.

Figure B. 6 :

 6 Figure B.6: Mean of Kendall-Tau for Linear (B.6a) and Circular (B.6b) Seriation for noisy observations of banded, Toeplitz, matrices, displayed for several scalings of the Laplacian embedding, with a fixed number of neighbors k = 15 and number of dimensions d = 10 in the d-LE.

Figure B. 7 :

 7 Figure B.7: Mean of Kendall-Tau for Linear (B.7a) and Circular (B.7b) Seriation for noisy observations of banded, Toeplitz, matrices, displayed for several scalings of the Laplacian embedding, with a fixed number of neighbors k = 15 and number of dimensions d = 20 in the d-LE.

Figure B. 8 :

 8 Figure B.8: Similarity matrix (with main diagonal removed) from synthetic, multiple chromosomes Hi-C data (DL1) without (B.8a) and with (B.8b) preprocessing as defined in Section 3.6.4.

  Figure B.10: Similarity matrices for the Spodoptera frugiperda data, Sf200 (B.10a), Sf669 (B.10c) and Sf846 (B.10e), and the corresponding orderings found with mdso (B.10b), (B.10d), (B.10f).

Figure D. 1 :Figure D. 2 : 1 Figure D. 3 :

 1213 Figure D.1: Original matrix S (with parameter =0.5) from which the data (A, c) is generated (A), output S out recovered from (A, c) by Algorithm 5.1 (used with ⌘-spectral) with N/n =1.33 (B) and with N/n =4(C). The meanDist metric is 0.98 for N/n =1 .33 (B) and 10.40 for N/n =4(C)

  vii D Supplementary Material for Chapter 5,S e r i a t i o nw i t hD u p l i c a t i o n s D.1 Supplementary Figures ................................ D.2 Supplementary Tables ................................

	viii

  Algorithm 1.2 Conditional gradient algorithm for constrained problem (1.5) Inputs: Initial point x 0 2 hull(C), target precision "

	for t =0,... do
	Solve linear minimization oracle
		s t = argmin s2hull(C)	hrf (x t ),si	(1.7)
	Get estimated gap
		∆ t = hx t s t , rf (x t )i	(1.8)
	if ∆ t  " then Stop end if Set	x t+1 = x t +	2 t +2	(s t x t )
	end for
	Output: x = x

t

The conditional gradient method (a.k.a Frank-Wolfe

[Frank and Wolfe, 1956

, Lacoste-Julien and Jaggi, 2015]), described in Algorithm 1.2, can circumvent this problem. It uses a linear minimization oracle (1.7) to produce a sequence of iterates that remain in hull(C) by construction. When f is convex and smooth, Algorithm 1.2 has a guaranteed convergence rate of O(1/t) towards a solution x ⇤ of (1.6). It also provides an estimated duality gap (1.8), due to an inequality involving the convexity of f .N o t et h a tu s i n gA l g o r i t h m1.2 is dependent upon the availability of an efficient linear minimization oracle to solve Equation (1.7). If this step can be performed efficiently (with a computationally cheap algorithm), then Algorithm 1.2 is likely to be efficient. The linear minimization oracle (LMO) depends solely on the set C. Indeed, in Equation (1.7), the gradient at x t is given, hence the problem is no other but a linear program on hull(C). For permutation problems, one can choose to represent permutations with vectors or matrices, resulting in two possible sets C.

  Definition 1.2.2. The convex hull of the set of n ⇥ n permutation matrices, cal led the Birkhoff polytope B n , is the set of all doubly-stochastic n ⇥ n matrices:

..,n, each integer i 2 [n] appear once, and only once. The convex hull of P is obtained by relaxing the integer constraints Π ij 2 {0, 1} into Π ij 2 [0, 1]. The resulting set, called the Birkhoff polytope, is defined as follows,

  [Lim and Wright, 2014], Definition 1.2.3. The permutahedron PH n ,t h ec o n v e xh u llo ft h es e to fp e r m u t a t i o nv e c t o r so f

size n,i s

Table 2 .

 2 1: Assembly results of the spectral method, compared to Miniasm, Canu and Racon, across the different datasets. For the spectral method, we give the results after contig merging (see §2.3.3); the number of contigs before this post-processing is given between parentheses. The best results in terms of average identity are highlighted in bold (but other metrics should also be used to compare the assemblies).

			Miniasm	Spectral	Canu	Miniasm+RaconMiniasm+Racon Spectral+Racon
							(2 iter.)
		Ref. size [bp]	3598621	3598621	3598621	3598621	3598621	3598621
	A. baylyi ONT R7.3 28x	Total size [bp] Ref. chr. [#] Contigs [#] Aln. ref [bp] Aln. query [bp] Misassemblies [#]	3531295 1 5 3445457(95.74%)3596249(99.93%)3595082(99.90%)3596858(99.95%)3596854(99.95%)3598181(99.99%) 3551582 3513432 3564823 3566438 3551094 1 1 1 1 1 1 (7) 1 5 5 1 (7) 3379002(95.69%)3549290(99.94%)3513081(99.99%)3564455(99.99%)3566021(99.99%)3550742(99.99%) 0 0 2 2 2 0
		Av g . i d e nt i ty	8 7 . 3 1	9 8 . 1 7	9 7 . 5 9	9 8 . 1 8	9 8 . 3 6	98.42
		Ref. size [bp]	4641652	4641652	4641652	4641652	4641652	4641652
		Total size [bp]	4759346	4662043	4625543	4647066	4643235	4629112
	E. coli	Ref. chr. [#]	1	1	1	1	1	1
	ONT	Contigs [#]	3	1 (4)	2	3	3	1 (4)
	R7.3	Aln. ref [bp]	4355121(93.83%)4612515(99.37%)4638255(99.93%)4640127(99.97%)4640127(99.97%)4641457(100.00%)
	30x	Aln. query [bp]	4432658(93.14%)4623823(99.18%)4625535(100.00%) 4642837(99.91%)4639816(99.93%)4628962(100.00%)
		Misassemblies [#]	0	2	8	3	3	2
		Av g . i d e nt i ty	8 9 . 2 8	9 8 . 8 0	9 9 . 4 0	9 9 . 3 1	9 9 . 4 5	99.46
		Ref. size [bp]	12157105	12157105	12157105	12157105	12157105	12157105
	S.	Total size [bp]	11813544	12213218	12142953	11926664	11926191	12167363
	cere-	Ref. chr. [#]	17	17	17	17	17	17
	visiae	Contigs [#]	29	71 (127)	36	29	29	71 (127)
	ONT	Aln. ref [bp]	11566318(95.14%) 12043050(99.06%) 12086977(99.42%) 12084923(99.41%) 12086556(99.42%) 12061384(99.21%)
	R7.3	Aln. query [bp]	11236806(95.12%) 12134480(99.36%) 12089056(99.56%) 11923058(99.97%) 11918621(99.94%) 12135284(99.74%)
	68x	Misassemblies [#]	0	7	34	18	19	11
		Av g . i d e nt i ty	8 9 . 0 0	9 8 . 0 0	9 8 . 3 3	9 8 . 4 9	98.63	98.61
		Ref. size [bp]	12157105	12157105	12157105	12157105	12157105	12157105
	S. cere-visiae ONT R9 86x	Total size [bp] Ref. chr. [#] Contigs [#] Aln. ref [bp] Aln. query [bp] Misassemblies [#]	11734150 17 30 11947453(98.28%) 11607131(95.48%) 12126980(99.75%) 12126663(99.75%) 12127467(99.76%) 11695983(96.21%) 11795644 12217497 12128279 12129086 11750114 17 17 17 17 17 48 (85) 26 30 29 48 (85) 11549494(98.43%) 11668882(98.93%) 12179843(99.69%) 12118506(99.92%) 12121202(99.93%) 11717047(99.72%) 0 23 39 18 19 36
		Av g . i d e nt i ty	9 3 . 5 5	9 8 . 8 1	9 9 . 0 2	9 9 . 1 6	99.20	99.10
		Ref. size [bp]	4641652	4641652	4641652	4641652	4641652	4641652
		Total size [bp]	4845211	4731239	4670125	4653228	4645420	4674460
	E. coli PacBio 161x	Ref. chr. [#] Contigs [#] Aln. ref [bp] Aln. query [bp]	1 1 4437473(95.60%)4617713(99.48%)4641652(100.00%) 4641551(100.00%) 4641500(100.00%) 4641652(100.00%) 1 1 1 1 1 2 (6) 1 1 1 2 (6) 4601587(94.97%)4705704(99.46%)4670125(100.00%) 4653140(100.00%) 4645420(100.00%) 4673065(99.97%)
		Misassemblies [#]	0	5	4	4	4	4
		Av g . i d e nt i ty	8 9 . 1 3	9 8 . 6 3	99.99	99.64	99.91	99.87
		Ref. size [bp]	12157105	12157105	12157105	12157105	12157105	12157105
	S. cere-visiae PacBio 127x	Total size [bp] Ref. chr. [#] Contigs [#] Aln. ref [bp] Aln. query [bp] Misassemblies [#]	12266420 17 30 11250453(92.54%) 11917823(98.03%) 12091868(99.46%) 12023040(98.90%) 12024968(98.91%) 12002816(98.73%) 12839034 12346258 12070971 12052148 12695031 17 17 17 17 17 90 (136) 29 30 30 90 (136) 11396172(92.91%) 12456415(97.02%) 12304982(99.67%) 12045088(99.79%) 12027812(99.80%) 12485128(98.35%) 0 57 76 61 59 68
		Av g . i d e nt i ty	8 8 . 2 9	9 8 . 4 1	99.87	99.43	99.72	99.54

Table 2 .

 2 2: Assembly of each chromosome of S. cerevisiae (for each chromosome, we used the subset of reads from the S. cerevisiae ONT R7.3 dataset that were mapped to it).

	Chr.	Ref size [bp] Contigs [#]	Aln. bp	Aln. bp	Misassem-	Avg.
				ref [bp]	query [bp]	-blies [#]	identity [%]
	I2 3 0 2 1 8	12 2 8 2 7 3 ( 9 9 . 1 6 % )	2 2 5 8 4 5 ( 9 8 . 4 3 % )	0	9 8 . 2 1
	II	813184	1	806340(99.16%)	797624(98.91%)	0	98.17
	III	316620	4	313707(99.08%)	326011(93.47%)	3	98.33
	IV	1531933	6	1519577(99.19%) 1539642(99.04%)	0	98.24
	V5 7 6 8 7 4	1	5 7 4 9 4 4 ( 9 9 . 6 7 % )	5 7 5 0 3 7 ( 9 9 . 3 0 % )3	9 8 . 3 7
	VI	270161	3	270161(100.00%)	285160(98.97%)	0	98.36
	VII	1090940	8	1088278(99.76%) 1115166(98.37%)	0	98.09
	VIII	562643	2	556839(98.97%)	561348(99.48%)	2	98.22
	IX	439888	2	437971(99.56%)	443785(97.81%)	0	98.38
	X7 4 5 7 5 1	2	7 4 0 6 9 6 ( 9 9 . 3 2 % )	7 3 8 8 5 9 ( 9 9 . 1 6 % )0	9 8 . 3 5
	XI	666816	2	665942(99.87%)	667003(99.46%)	0	98.35
	XII	1078177	5	1067559(99.02%) 1084233(98.50%)	2	98.27
	XIII	924431	4	922948(99.84%)	937417(99.58%)	1	98.12
	XIV	784333	2	779066(99.33%)	783072(99.35%)	0	98.41
	XV	1091291	3	1089941(99.88%) 1088832(99.49%)	0	98.34
	XVI	948066	11	942078(99.37%)	1015108(97.50%)	1	97.83
	Chrmt.	85779	5	65196(76.00%)	69107(80.98%)	-	90.32

  Contents 3.1 Introduction .................................... 3.2 Related Work ....................................3.2.1 Spectral Ordering for Linear Seriation .................. 3.2.2 Laplacian Embedding ........................... 3.2.3 Link with Continuous Operators ..................... 3.2.4 Other embeddings ............................. 3.2.5 Ordering points lying on a curve ..................... 3.3 Spectral properties of some (circular) Robinson matrices ........... 3.3.1 Circular Seriation with Symmetric, Circulant matrices .........

3.3.2 (Linear) Robinson Toeplitz matrices ................... 3.3.3 Spectral properties of the Laplacian ................... 3.4 Recovering Ordering on Filamentary Structure ................. 3.4.1 The Algorithm ............................... 3.4.2 Illustration of Algorithm 3.3 ........................ 3.5 Perturbation analysis ................................ 3.5.1 Application of the Davis-Kahan Theorem ................ 3.5.2 Exact recovery with noise for Algorithm 3.2 ............... 3.6 Numerical Results ................................. 3.6.1 Synthetic Experiments ........................... 3.6.2 Genome assembly experiment : bacterial genomes with ONT long-reads 3.6.3 Genome assembly using Hi-C data .................... 3.6.4 Assembly of genomes with multiple chromosomes with Hi-C data ... 3.6.5 Finding circular orderings with single-cell Hi-C data .......... 3.7 Conclusion .....................................

  such A, Seriation seeks to recover this permutation Π,[START_REF] Von | A tutorial on spectral clustering[END_REF] to the case where we wish to infer a latent ordering rather than a latent clustering on the data. Roughly speaking, both methods embed the elements on a line and associate a coordinate f i 2 R to each element i 2 [n]. Spectral clustering addresses a graph-cut problem by grouping these coordinates into two clusters. Spectral ordering[Atkins et al., 1998] addresses Linear Seriation by sorting the f i . The clustering method is closely related to the ordering, as noted in[START_REF] Ding | Linearized cluster assignment via spectral ordering[END_REF].

	find Π 2 P find Π 2 P	such that ΠAΠ T 2 L R such that ΠAΠ T 2 C R	(Linear Seriation) (Circular Seriation)
	A widely used method for Linear Seriation is the spectral relaxation presented in Chap-
	ter 1, 1.2.2, based on the graph Laplacian of the similarity matrix. It transposes Spectral
	Clustering [

A graph-cut partitions the data in two clusters. When seeking to cluster data in K groups with K>2, one can recursively iterate graph cuts in the sub-groups obtained at the previous iterations. However, most Spectral Clustering algorithms actually use a Laplacian embedding of dimension d>1, denoted d-LE in the following, in order to find K clusters. Latent cluster structure is assumed to be enhanced in the d-LE, and the k-means algorithm

[MacQueen et al., 1967[START_REF] Hastie | Unsupervised learning[END_REF]

seamlessly identifies the clusters from the embedding. In contrast, Spectral Ordering is restricted to d =1by the sorting step (there is no total order relation on R d for d>1). Still, the latent linear structure may emerge from the d-LE,i ft h ep o i n t sa r e distributed along a curve. Also, for d =2 , it may capture the circular structure of the data and allow for solving Circular Seriation. One must then recover a (circular) ordering of points lying in a 1D manifold (a curve, or filament) embedded in R d .

  circular Toeplitz, from Theorem 3.3.2,t h er o wo fV are n ordered points uniformly sampled on the unit circle. Because applying a rotation is equivalent to translating the angle of these points on the circle. It follows that there exists a

	cyclic permutation ⌧ such that
	sup i=1,...,n	||p i p⌧(i) || 2 	2 3/2 min( min(| 1 |, | 2 p 2||L H || 2 , ||L H || F ) 1 |)	,
	3.5.2 Exact recovery with noise for Algorithm 3.2

  Denote by (p i ) i=1,...,n and (p i ) i=1,...,n the 2-LE coming from L and L respectively. (p 1 (i) ) i=1,...,n is the 2-LE coming from the graph Laplacian of Π T ÃΠ.

Applying Proposition 3.5.2 with Π T ÃΠ, there exists a cyclic permutation such that sup i=1,...,n ||p 1

Table 3 .

 3 1: Seriation results on synthetic Hi-C data from linear single-stranded genome.

			Spectral mdso
	N = 100	⌧ (%) Time (s)	99.4 0.74	99.7 0.18
	N = 150	⌧ (%) Time (s)	99.7 0.093	100 0.23
	N = 300	⌧ (%) Time (s)	100 0.42	100 0.70
	N = 1000	⌧ (%) Time (s)	100 2.1	100 1.7
	were wrongfully assembled together by the assembler. The respective lengths of the resulting
	matrices Sf200, Sf669 and Sf846 are 198, 284, and 461 bins.

We acknowledge Dominique Lavenier from the GenOuest group at INRIA Rennes for providing the synthetic and Plasmodium knowlesi data, and Fabrice Legeai, from the same group, for the Spodoptera frugiperda data.

Table 3 .

 3 2: Seriation results on synthetic Hi-C data from genomes with multiple chromosomes.

			SC+SO SC+mdso mdso tSNE-mdso
		# Chr.	7	7	7	7
	DL1	Purity Cluster dist.	100 0.0	100 0.0	100 0.0	100 0.0
		⌧ (%)	96.7	83.3	96.7	95.6
		Time (s)	0.079	0.24	0.31	3.77
		# Chr.	7	7	7	7
	DL2	Purity	100	100	100	100
		Cluster dist.	0.0	0.0	0.0	0.0
		⌧ (%)	99.5	99.2	99.4	99.5
		Time (s)	0.19	0.53	1.45	15.9
	⇡

2 (i) < ⇡ 2 (j),o r⇡ 1 (i) < ⇡ 1 (j) and ⇡ 2 (i) > ⇡ 2 (j).

Then, the Kendall-Tau rank correlation is defined by, ⌧ = number of concordant pairs number of discordant pairs n(n 1)/2 . (3.16)

Table 3 .

 3 3: Seriation results on real Hi-C data from the Plasmodium knowlesi genome

	SC+SO SC+mdso mdso tSNE-mdso
	# Chr.	14	14	8	16.7 ± 0.5
	Purity	76.3	76.3	-	-
	Cluster dist.	3.81	3.81	6.61 2.83 ± 0.23
	⌧ (%)	77.3	69.6	18.9 85.0 ± 11.7
	Time (s)	0.30	1.45	7.76	43.9 ± 2.8
	(a) tSNE embedding			(b) sub-orderings

Table 3 .

 3 4: Seriation results on real Hi-C data from a Spodoptera frugiperda genome.

			Spectral mdso tSNE-mdso
		# Chr.	1	2	4.87 ± 0.7
	Sf200	⌧ (%)	92.7	91.6	86.4 ± 3.0
		Time (s)	0.033	0.14 3.06 ± 0.12
		# Chr.	1	4	5.9 ± 0.5
	Sf669	⌧ (%)	75.7	88.2	87.8 ± 2.3
		Time (s)	0.049	0.19 4.64 ± 0.07
		# Chr.	1	6	12.2 ± 0.45
	Sf846	⌧ (%)	95.8	86.7	86.8 ± 0.6
		Time (s)	0.063	0.22 8.03 ± 0.66

the previous subsection. Rather, given a set of n individual cells, Hi-C frequency interaction

Table 3 .

 3

	CROC	G1	E-S	M-S L-S/G2	avg
	MDS	0.938 0.966 0.917	0.917	0.936
	Spec	0.932 0.951 0.922	0.886	0.923
	Mdso 5	0.943 0.964 0.927	0.914	0.937
	Mdso 10 0.943 0.967 0.921	0.905	0.934
	Table 3.5 shows a comparison of the CROC score according to the embedding use. MDS
	and Spec performs similarly. Both embeddings (MDSO) resulting from Algorithm 3.3 lead to
	a better score with respect to the the baseline spectral Algorithm 3.2. Nevertheless there is no
	outstanding benefit in using the processing of Algorithm 3.3 instead of a simple MDS on that
	particular type of data. Figure 3.11 illustrates the MDS and spectral-Laplacian embeddings

5:

Comparison of CROC scores between MDS embedding, basic spectral embedding (Spec) and the pre-processing of our Multi-dimensional Spectral ordering method (Mdso) depending on the neighborhood parameter k. The score are only slighty better.

  We say that A 2 S n is a strong-R-matrix (or strong Robinson matrix) iff it is symmetric and satisfies A ij  A kl for all (i, j, k, l) such that |i j| > |k l|.

	, to bioinformatics [Atkins
	and Middendorf, 1996, Cheema et al., 2010, Jones et al., 2012], the one of interest throughout
	this manuscript is genome assembly. We have introduced the Robinson structural hypothesis
	on similarity matrices underpinning Seriation in Chapters 1 and 3, in Definition 1.1.1. Here,
	we consider a stronger assumption, introduced below.
	Definition 4.1.1.

  Remark that the search space P n is discrete and of cardinality n!, thus preventing the use of exhaustive search or greedy branch and bound methods for Seriation or 2-SUM when n gets large[Hahsler et al., 2008]. Yet, for pre-R ⇤ matrices, Seriation is equivalent to 2-SUM[START_REF] Fogel | Aspremont. Convex relaxations for permutation problems[END_REF], which can be solved exactly in polynomial time, using the spectral relaxation from

	torial problem over permutations, written	.1. Given A 2 S n , 2-SUM is an combina-
	minimize such that ⇡ 2 P n P n i,j=1 A ij |⇡ i ⇡ j | 2	(2-SUM)
	Atkins et al. [1998], presented in Chapter 1 (Algorithm 1.1), and exploited in Chapter 2 in the
	context of genome assembly.
	Problem 2-SUM is also a particular case of the Quadratic Assignment Problem [Koopmans
	and Beckmann, 1957], written
		n X
	min ⇡2Pn	i,j=1

  1} for |i j| = k +1, with the integer k +1 denoting the bandwidth of S. Given S Π and the corresponding k,t h e distance between ΠAΠ T and S Π appearing in Robust Seriation is separable (whether we use the l 1 or Frobenius norm, since A 2 {0, 1} n⇥n ) and can be grouped into three terms, according to whether (i, j) is such that |i j| >k+1, |i j|  k or |i j| = k +1. The first term, n out (k) 0, equals the number of non-zero elements of ΠAΠ T such that |i j| >k+1.T h e second, n in (k) 0, equals the number of zero elements of ΠAΠ

	n in (k)	2(n	1) s	(n	1)

T such that |i j|  k. The third equals zero, because setting the (k+1)-th diagonal of S identical to the (k+1)-th diagonal of ΠAΠ T does not violate the R property of S Π ,a n dS Π is by definition the strong-R-matrix that minimizes the distance to ΠAΠ T .F o ra n yΠ,i fk> , the number of non-zeros elements inside the band of width k being bounded by the number of non-zero elements of A,w eh a v e s. Similarly, for k  , n out (k) s.F o rΠ = I, as long as k  , n out (k)  s decreases with k and n in

  4.1 Conditional gradient algorithm for permutation problem min x2hull(P) f (x).(hull(P) can be either B or PH). Initial point x 0 2 hull(P), target precision "

	Inputs: for t =0,... do
	Solve linear minimization oracle
		s t = argmin s2P	hrf (x t ),si	(4.2)
	Get estimated gap
		∆ t = hx t s t , rf (x t )i	(4.3)
	if ∆ t  " then Stop end if Set	x t+1 = x t +	2 t +2	(s t x t )
	end for
	Output: x = x

t The latter, denoted PHCD [Lim and Wright, 2016] in the following, uses the sorting-network based representation of permutation vectors of Goemans [2014] and performs coordinate descent in the convex hull of permutation vectors PH.

  ±0.08 7.19 ±0.31 2.46 ±0.14 67.6 ±4.1 7.4 ±0.2 3.37e+00 LWC D 0 . 8 9 ±0.08 7.18 ±0.30 2.46 ±0.14 67.5 ±3.9 7.4 ±0.2 2.99e+00 UBI 0.89 ±0.06 7.32 ±0.31 2.52 ±0.12 69.5 ±3.3 7.5 ±0.2 1.45e+00 Manopt 0.86 ±0.06 7.72 ±0.58 2.66 ±0.18 73.2 ±5.2 7.6 ±0.4 3.90e+00 ⌘-Spectral 0.97 ±0.00 6.74 ±0.13 2.03 ±0.02 50.8 ±0.8 7.6 ±0.2 1.07e+00 HGnCR 0.89 ±0.22 6.91 ±0.52 2.11 ±0.26 53.6 ±8.6 7.7 ±0.4 9.06e+00 H-FAQ 0.95 ±0.08 6.84 ±0.32 2.01 ±0.08 49.0 ±3.9 7.7 ±0.3 4.28e-01 H-LWCD 0.94 ±0.09 6.88 ±0.34 2.03 ±0.11 49.7 ±5.0 7.7 ±0.3 3.00e+00 H-UBI 0.97 ±0.00 6.74 ±0.13 2.05 ±0.02 51.4 ±1.1 7.6 ±0.2 3.08e+00 H-Manopt 0.92 ±0.06 7.05 ±0.39 2.26 ±0.15 59.7 ±5.2 7.6 ±0.3 9.22e+00 ±0.09 7.03 ±0.42 2.01 ±0.09 46.0 ±4.8 8.0 ±0.4 3.32e+00

		⌧	-	×1e-6 Huber	×1e-6 R2SUM	Dist2R	×1e-6 2SUM	Time (s)
	spectral	0.86 ±0.06 7.76 ±0.61 2.67 ±0.19 73.6 ±5.3 7.7 ±0.4	3.54e-01
	GnCR	0.87 ±0.15 7.21 ±0.40 2.47 ±0.17 67.6 ±4.7 7.5 ±0.3	6.99e-01
	FAQ 0 . 8 9 R-FAQ 0.95 ±0.10 6.97 ±0.40 1.99 ±0.08	44.9 ±4.3	7.9 ±0.4	3.39e-01
	R-LWCD	0.94

Table 4 .

 4 2: Kendall-⌧ score for different values of s/s lim , for the same methods as in Table 4.1, and n = 200, = 20. s/s lim =0.5 s/s lim =1 s/s lim =2.5 s/s lim =5 s/s lim =7.5 s/s lim = 10

	spectral	0.96 ±0.01	0.95 ±0.01	0.91 ±0.03	0.86 ±0.06	0.84 ±0.06	0.80 ±0.09
	GnCR	0.98 ±0.00	0.96 ±0.04	0.93 ±0.07	0.87 ±0.15	0.81 ±0.20	0.80 ±0.18
	FAQ	0 . 9 8 ±0.00	0.97 ±0.00	0.94 ±0.02	0.89 ±0.08	0.87 ±0.08	0.82 ±0.13
	LWC D	0 . 9 8 ±0.00	0.97 ±0.00	0.94 ±0.02	0.89 ±0.08	0.87 ±0.08	0.82 ±0.13
	UBI	0.97 ±0.00	0.96 ±0.01	0.92 ±0.03	0.89 ±0.06	0.86 ±0.07	0.82 ±0.12
	Manopt	0.97 ±0.00	0.95 ±0.01	0.91 ±0.03	0.86 ±0.06	0.84 ±0.06	0.80 ±0.09
	⌘-Spectral	0.99 ±0.00	0.99 ±0.00	0.98 ±0.00	0.97 ±0.00	0.96 ±0.00	0.94 ±0.06
	HGnCR	1.00 ±0.00	0.99 ±0.00	0.99 ±0.00	0.89 ±0.22	0.85 ±0.23	0.83 ±0.25
	H-FAQ	1.00 ±0.00	1.00 ±0.00	0.99 ±0.01	0.95 ±0.08	0.94 ±0.09	0.91 ±0.13
	H-LWCD	1.00 ±0.00	1.00 ±0.00	0.99 ±0.02	0.94 ±0.09	0.94 ±0.09	0.90 ±0.14
	H-UBI	0.99 ±0.00	0.99 ±0.00	0.98 ±0.00	0.97 ±0.00	0.96 ±0.01	0.94 ±0.03
	H-Manopt	1.00 ±0.00	0.99 ±0.00	0.97 ±0.02	0.92 ±0.06	0.89 ±0.07	0.84 ±0.10
	R-FAQ	1.00 ±0.00	1.00 ±0.00	0.99 ±0.04	0.95 ±0.10	0.94 ±0.10	0.90 ±0.15
	R-LWCD	0.99 ±0.00	1.00 ±0.00	0.99 ±0.04	0.94 ±0.09	0.94 ±0.10	0.90 ±0.16

Table 4 .

 4 3: Kendall-⌧ score for different values of s/s lim , for the spectral method and Frank-Wolfe with default and initialized tie-breaks (-I variants), with n = 200, = 20. s/s lim =0.5 s/s lim =1 s/s lim =2.5 s/s lim =5 s/s lim =7.5 s/s lim = 10

	spectral	0.96 ±0.01	0.95 ±0.01	0.91 ±0.03	0.86 ±0.06	0.84 ±0.06	0.80 ±0.09
	FWTB	0.40 ±0.27	0.33 ±0.27	0.32 ±0.26	0.34 ±0.22	0.28 ±0.21	0.24 ±0.20
	H-FWTB	0.50 ±0.31	0.36 ±0.27	0.32 ±0.25	0.32 ±0.22	0.25 ±0.21	0.22 ±0.18
	FWTB-i	0.91 ±0.20	0.92 ±0.13	0.84 ±0.19	0.73 ±0.20	0.71 ±0.13	0.64 ±0.15
	H-FWTB-i	0.98 ±0.01	0.94 ±0.13	0.86 ±0.15	0.70 ±0.18	0.63 ±0.15	0.57 ±0.17

Table 4 .

 4 4: Seriation results on real Hi-C data from the Plasmodium knowlesi genome.

		SC+SO SC+mdso SC+⌘-SO mdso tSNE-mdso
	# Chr.	14	14	14	8	16.7 ± 0.5
	Purity	76.3	76.3	76.3	-	-
	Cluster dist.	3.81	3.81	3.81	6.61 2.83 ± 0.23
	⌧ (%)	77.3	69.6	79.3	18.9 85.0 ± 11.7
	Time (s)	0.30	1.45	2.04	7.76	43.9 ± 2.8

Table 4 .

 4 5: Seriation results on real Hi-C data from a Spodoptera frugiperda genome.

			Spectral ⌘-Spectral mdso
		# Chr.	1	1	2
	Sf200	⌧ (%)	92.7	95.9	91.6
		Time (s)	0.033	0.026	0.14
		# Chr.	1	1	4
	Sf669	⌧ (%)	75.7	75.9	88.2
		Time (s)	0.049	0.61	0.19
		# Chr.	1	1	6
	Sf846	⌧ (%)	95.8	97.7	86.7
		Time (s)	0.063	0.9	0.22

Table 5 .

 5 1: Results of synthetic experiments for Seriation with Duplications from matrices S 2 M N ( ,s) with n = 200,

Table 5 .

 5 2: Results of synthetic experiments for Seriation with Duplications from dense, strong-R matrices of size n = 200, with the same metrics and methods as in Table5.1, with =0.5.

	N/n	method	d2S	Huber	meanDist stdDist	Time
				(×1e-7)			(×1e-2s)
		spectral	0.25 ±0.04 1.36 ±0.03	6.1 ±1.8	7.9 ±1.6	8.74 ±4.85
	1.33	⌘-Spectral 0.15 ±0.02 1.30 ±0.01	2.2 ±0.7	3.7 ±1.1	6.12 ±4.84
		H-UBI	0.24 ±0.04 1.35 ±0.03	5.5 ±1.6	7.3 ±1.4 11.06 ±7.56
		spectral	0.27 ±0.02 1.41 ±0.02	9.5 ±1.6	8.4 ±1.3	7.47 ±3.20
	2	⌘-Spectral 0.22 ±0.02 1.37 ±0.02	6.6 ±1.5	6.7 ±1.9	7.89 ±3.89
		H-UBI	0.26 ±0.02 1.40 ±0.02	9.0 ±1.5	8.1 ±1.2 10.09 ±4.90
		spectral	0.18 ±0.01 1.35 ±0.01	14.4 ±2.8	8.7 ±2.7	6.53 ±1.90
	4	⌘-Spectral 0.18 ±0.01 1.35 ±0.01	14.3 ±2.9	8.9 ±2.9	7.59 ±2.28
		H-UBI	0.19 ±0.01 1.35 ±0.01	14.8 ±2.5	8.8 ±2.1	8.62 ±2.46

Table 5 .

 5 3: Results of synthetic experiments for Seriation+Clustering with Duplications from dense, strong-R matrices of size n = 200, with an additive block matrix with 5 clusters, with Algorithm 5.1 (that do not take the cluster structure into account), denoted SerDupli and Algorithm 5.3, denoted SerDuClus. Both are used with the ⌘-Spectral method at step 3 of the alternate projections Algorithm. The results are averaged over 20 experiments and the standard deviation is given after the ± sign.

	N/n	method	Huber	meanDist stdDist
	1.33	SerDupli SerDuClus 9.265e+06 ±2.439e+06 1.266e+07 ±4.946e+06 23.9 ±16.2 19.9 ±12.2 9.04 ±10.4 8.9 ±8.1
	2	SerDupli SerDuClus 1.373e+07 ±4.284e+06 1.683e+07 ±5.283e+06	38.8 ±9.9 30.6 ±12.8	26.6 ±7.1 20.2 ±8.1
	4	SerDupli SerDuClus 2.512e+07 ±6.704e+06 3.639e+07 ±4.357e+06 42.0 ±11.0 35.5 ±9.1	18.5 ±4.9 15.5 ±5.0

Table A .

 A 1: Running time for the different methods on the datasets presented in Section 2.3.1 (Chapter 2)

				Spectral	Spectral	Canu	Minimap	Racon	Racon
				Layout	(full,		+	after	after
					+Min-		Miniasm	Miniasm	Spectral
					imap)
	A.	baylyi	Runtime	0:00:23	0:12:52	0:25:55	0:00:28	0:01:54	0:01:48
	ONT R7.3	[h:mm:ss]	(0:00:59)
	28x		Max mem	1.966	1.966	3.827	1.499	0.756	0.484
			[Gb]
	E.	coli	Runtime	0:00:41	0:16:15	0:28:40	0:00:13	0:04:36	0:02:14
	ONT R7.3	[h:mm:ss]	(0:01:25)
	30x		Max mem	1.216	1.216	4.655	2.099	0.879	0.645
			[Gb]
	S.	cere-	Runtime	0:01:41	1:41:20	4:33:08	0:01:17	0:21:11	0:21:32
	visiae ONT	[h:mm:ss]	(0:07:60)
	R7.3 68x	Max mem	12.208	12.208	4.015	8.506	2.376	2.325
			[Gb]
	S.	cere-	Runtime	0:03:38	2:26:44	7:15:41	0:02:14	0:23:09	0:22:03
	visiae ONT	[h:mm:ss]	(0:09:28)
	R9 86x	Max mem	32.928	32.928	3.986	12.397	2.966	2.775
			[Gb]
	E.	coli	Runtime	0:05:19	1:32:13	0:51:32	0:01:16	0:16:51	0:18:18
	PacBio	[h:mm:ss]	(0:05:44)
	161x		Max mem	21.650	21.650	3.770	9.969	8.082	4.619
			[Gb]
	S.	cere-	Runtime	0:03:11	2:59:41	1:50:23	0:02:10	0:20:54	0:23:32
	visiae		[h:mm:ss]	(0:07:01)
	PacBio	Max mem	32.184	32.184	3.810	16.881	4.290	4.307
	127x		[Gb]

to the backbone sequence with Minimap and to run Racon for the consensus (Racon requires a backbone sequence, obtained either with Miniasm or Spectral in the present experiments).

Table A .

 A 2: Assembly results of several assemblers across the datasets corrected with Canu

			Miniasm	Spectral	Canu	Miniasm+RaconMiniasm+Racon Spectral+Racon
							(2 iter.)
		Ref. size [bp]	3598621	3598621	3598621	3598621	3598621	3598621
	A.	Total bases [bp]	3493724	3523055	3516777	3540178	3540766	3522315
	baylyi	Ref. chr. [#]	1	1	1	1	1	1
	ONT	Contigs [#]	5	2 (9)	2	5	5	2 (9)
	R7.3	Aln. bp ref [bp]	3594663(99.89%)3596069(99.93%)3595264(99.91%)3595193(99.90%)3595193(99.90%)3596269(99.93%)
	28x	Aln. bp query [bp] 3492976(99.98%)3522804(99.99%)3516440(99.99%)3539856(99.99%)3540444(99.99%)3522311(100.00%)
	(26x)	Misassemblies [#]	2	1	2	2	2	1
		Av g . i d e nt i ty	9 6 . 4 0	97.87	97.61	97.79	97.85	97.86
		Ref. size [bp]	4641652	4641652	4641652	4641652	4641652	4641652
	E. coli ONT R7.3 30x (27x)	Total bases [bp] Ref. chr. [#] Contigs [#] Aln. bp ref [bp] Aln. bp query [bp] 4597389(100.00%) 4613972(100.00%) 4627577(100.00%) 4617119(100.00%) 4617099(100.00%) 4613520(100.00%) 4597538 4613973 4627578 4617120 4617100 4613521 1 1 1 1 1 1 3 1 (8) 2 3 3 1 (8) 4639179(99.95%)4639815(99.96%)4639396(99.95%)4639355(99.95%)4639355(99.95%)4639420(99.95%) Misassemblies [#] 2 2 4 2 2 2
		Av g . i d e nt i ty	9 8 . 8 9	99.43	99.41	99.42	99.43	99.43
	S. cere-visiae ONT R7.3 68x (38x)	Ref. size [bp] Total bases [bp] Ref. chr. [#] Contigs [#] Aln. bp ref [bp] Aln. bp query [bp] 11814252(100.00%) 11930637(99.76%) 12069253(99.65%) 11876268(99.99%) 11876225(99.99%) 11925068(99.79%) 12157105 12157105 12157105 12157105 12157105 12157105 11814836 11959669 12112186 11877015 11876882 11949674 17 17 17 17 17 17 29 67 (126) 37 28 28 67 (126) 12061456(99.21%) 11963869(98.41%) 12068379(99.27%) 12062161(99.22%) 12061809(99.22%) 11969742(98.46%) Misassemblies [#] 19 22 26 20 20 24 Av g . i d e nt i ty 9 7 . 8 1 9 8 . 3 2 9 8 . 3 6 98.39 98.39 98.38
		Ref. size [bp]	12157105	12157105	12157105	12157105	12157105	12157105
	S.	Total bases [bp]	11946760	12081487	12184545	11970672	11970529	12061759
	cere-	Ref. chr. [#]	17	17	17	17	17	17
	visiae	Contigs [#]	21	65 (108)	30	20	20	65 (108)
	ONT	Aln. bp ref [bp]	12055448(99.16%) 11851023(97.48%) 12110461(99.62%) 12056562(99.17%) 12056734(99.17%) 11879607(97.72%)
	R9 86x	Aln. bp query [bp] 11944969(99.99%) 12043650(99.69%) 12184122(100.00%) 11970041(99.99%) 11969729(99.99%) 12040521(99.82%)
	(40x)	Misassemblies [#]	21	32	26	22	22	38
		Av g . i d e nt i ty	9 8 . 8 3	9 8 . 9 0	99.06	99.06	99.05	99.04
		Ref. size [bp]	4641652	4641652	4641652	4641652	4641652	4641652
		Total bases [bp]	4642736	4663427	4670125	4642423	4642443	4662179
	E. coli	Ref. chr. [#]
	PacBio
	161x
	(38x)

Table C .

 C 1: Kendall-⌧ score for different values of s/s lim , for the same methods as in Table 4.1, for different values of n (namely, 100 , 200 , 500), and = n/10 (namely, 10 , 20 , 50). s/s lim =0.5 s/s lim =1 s/s lim =2.5 s/s lim =5 s/s lim =7.5 s/s lim = 10 Table C.2: Kendall-⌧ score for different values of s/s lim , for the same methods as in Table 4.1, for different values of n (namely, 100 , 200 , 500), and = n/20 (namely, 5 , 10 , 25). s/s lim =0.5 s/s lim =1 s/s lim =2.5 s/s lim =5 s/s lim =7.5 s/s lim = 10

		spectral spectral	0.91 ±0.08 0.46 ±0.24	0.83 ±0.13 0.39 ±0.21	0.72 ±0.19 0.31 ±0.20	0.62 ±0.21 0.25 ±0.16	0.55 ±0.20 0.22 ±0.15	0.48 ±0.21 0.20 ±0.14
		GnCR GnCR	0.92 ±0.13 0.43 ±0.28	0.82 ±0.23 0.37 ±0.21	0.70 ±0.26 0.32 ±0.21	0.62 ±0.26 0.25 ±0.16	0.55 ±0.25 0.25 ±0.14	0.48 ±0.24 0.20 ±0.13
	n = 100 n = 100	FAQ LWC D FAQ LWC D	0 . 9 3 ±0.09 0 . 9 3 ±0.10 0 . 4 5 ±0.25 0 . 4 5 ±0.26	0.85 ±0.17 0.85 ±0.17 0.39 ±0.22 0.39 ±0.22	0.72 ±0.24 0.72 ±0.24 0.31 ±0.21 0.31 ±0.21	0.61 ±0.25 0.61 ±0.25 0.25 ±0.17 0.25 ±0.17	0.55 ±0.25 0.55 ±0.25 0.23 ±0.15 0.23 ±0.15	0.48 ±0.23 0.48 ±0.23 0.22 ±0.14 0.22 ±0.14
		UBI UBI	0.92 ±0.09 0.45 ±0.26	0.85 ±0.16 0.40 ±0.22	0.73 ±0.24 0.32 ±0.21	0.62 ±0.24 0.26 ±0.17	0.56 ±0.24 0.23 ±0.15	0.49 ±0.23 0.23 ±0.14
		Manopt Manopt	0.92 ±0.08 0.46 ±0.25	0.84 ±0.13 0.40 ±0.21	0.72 ±0.19 0.31 ±0.20	0.62 ±0.21 0.25 ±0.16	0.55 ±0.20 0.22 ±0.15	0.48 ±0.21 0.21 ±0.14
		⌘-Spectr. ⌘-Spectr.	0.99 ±0.00 0.65 ±0.33	0.98 ±0.00 0.50 ±0.28	0.89 ±0.17 0.37 ±0.24	0.74 ±0.25 0.28 ±0.19	0.65 ±0.26 0.25 ±0.16	0.56 ±0.26 0.23 ±0.16
		HGnCR HGnCR	0.98 ±0.06 0.53 ±0.31	0.96 ±0.14 0.43 ±0.26	0.80 ±0.25 0.36 ±0.22	0.65 ±0.30 0.25 ±0.17	0.54 ±0.29 0.22 ±0.15	0.49 ±0.29 0.18 ±0.14
	n = 100 n = 100	H-FAQ H-LWCD H-FAQ H-LWCD	0.97 ±0.09 0.97 ±0.09 0.48 ±0.26 0.49 ±0.27	0.90 ±0.16 0.90 ±0.16 0.41 ±0.23 0.42 ±0.23	0.80 ±0.25 0.80 ±0.25 0.33 ±0.23 0.34 ±0.23	0.70 ±0.29 0.70 ±0.29 0.28 ±0.17 0.28 ±0.18	0.64 ±0.28 0.65 ±0.28 0.24 ±0.16 0.24 ±0.16	0.55 ±0.26 0.55 ±0.28 0.23 ±0.15 0.23 ±0.16
		H-UBI H-UBI	0.99 ±0.00 0.60 ±0.35	0.98 ±0.04 0.52 ±0.29	0.88 ±0.20 0.40 ±0.26	0.75 ±0.25 0.28 ±0.19	0.62 ±0.26 0.25 ±0.16	0.54 ±0.25 0.23 ±0.15
		H-Manopt H-Manopt	0.98 ±0.05 0.54 ±0.30	0.91 ±0.14 0.44 ±0.25	0.78 ±0.23 0.33 ±0.22	0.65 ±0.24 0.25 ±0.16	0.56 ±0.21 0.22 ±0.15	0.48 ±0.21 0.21 ±0.14
	n = 100 n = 100	R-FAQ R-LWCD R-FAQ R-LWCD	0.96 ±0.09 0.95 ±0.09 0.48 ±0.25 0.47 ±0.24	0.91 ±0.16 0.89 ±0.17 0.41 ±0.22 0.41 ±0.22	0.80 ±0.25 0.78 ±0.24 0.33 ±0.21 0.32 ±0.21	0.70 ±0.28 0.69 ±0.28 0.26 ±0.18 0.25 ±0.16	0.65 ±0.27 0.62 ±0.28 0.23 ±0.15 0.22 ±0.15	0.54 ±0.28 0.53 ±0.28 0.23 ±0.16 0.22 ±0.15
		spectral spectral	0.96 ±0.01 0.72 ±0.21	0.95 ±0.01 0.59 ±0.24	0.91 ±0.03 0.49 ±0.26	0.86 ±0.06 0.42 ±0.23	0.84 ±0.06 0.35 ±0.20	0.80 ±0.09 0.31 ±0.18
		GnCR GnCR	0.98 ±0.00 0.69 ±0.29	0.96 ±0.04 0.56 ±0.31	0.93 ±0.07 0.45 ±0.26	0.87 ±0.15 0.37 ±0.27	0.81 ±0.20 0.34 ±0.22	0.80 ±0.18 0.32 ±0.23
	n = 200 n = 200	FAQ LWC D FAQ LWC D	0 . 9 8 ±0.00 0 . 9 8 ±0.00 0 . 7 2 ±0.24 0 . 7 2 ±0.24	0.97 ±0.00 0.97 ±0.00 0.60 ±0.26 0.60 ±0.26	0.94 ±0.02 0.94 ±0.02 0.49 ±0.26 0.49 ±0.27	0.89 ±0.08 0.89 ±0.08 0.41 ±0.24 0.42 ±0.25	0.87 ±0.08 0.87 ±0.08 0.35 ±0.21 0.36 ±0.21	0.82 ±0.13 0.82 ±0.13 0.33 ±0.20 0.33 ±0.20
		UBI UBI	0.97 ±0.00 0.73 ±0.26	0.96 ±0.01 0.59 ±0.28	0.92 ±0.03 0.50 ±0.28	0.89 ±0.06 0.42 ±0.25	0.86 ±0.07 0.35 ±0.21	0.82 ±0.12 0.33 ±0.21
		Manopt Manopt	0.97 ±0.00 0.72 ±0.22	0.95 ±0.01 0.59 ±0.24	0.91 ±0.03 0.49 ±0.26	0.86 ±0.06 0.42 ±0.24	0.84 ±0.06 0.35 ±0.20	0.80 ±0.09 0.31 ±0.18
		⌘-Spectr. ⌘-Spectr.	0.99 ±0.00 0.99 ±0.00	0.99 ±0.00 0.91 ±0.21	0.98 ±0.00 0.65 ±0.33	0.97 ±0.00 0.52 ±0.30	0.96 ±0.00 0.41 ±0.25	0.94 ±0.06 0.37 ±0.23
		HGnCR HGnCR	1.00 ±0.00 0.73 ±0.33	0.99 ±0.00 0.61 ±0.32	0.99 ±0.00 0.50 ±0.31	0.89 ±0.22 0.44 ±0.29	0.85 ±0.23 0.39 ±0.25	0.83 ±0.25 0.35 ±0.22
	n = 200 n = 200	H-FAQ H-LWCD H-FAQ H-LWCD	1.00 ±0.00 1.00 ±0.00 0.75 ±0.24 0.75 ±0.23	1.00 ±0.00 1.00 ±0.00 0.63 ±0.27 0.62 ±0.27	0.99 ±0.01 0.99 ±0.02 0.53 ±0.29 0.53 ±0.29	0.95 ±0.08 0.94 ±0.09 0.46 ±0.27 0.46 ±0.27	0.94 ±0.09 0.94 ±0.09 0.38 ±0.23 0.38 ±0.23	0.91 ±0.13 0.90 ±0.14 0.35 ±0.23 0.35 ±0.22
		H-UBI H-UBI	0.99 ±0.00 0.94 ±0.19	0.99 ±0.00 0.82 ±0.30	0.98 ±0.00 0.69 ±0.34	0.97 ±0.00 0.57 ±0.32	0.96 ±0.01 0.46 ±0.28	0.94 ±0.03 0.40 ±0.23
		H-Manopt H-Manopt	1.00 ±0.00 0.84 ±0.23	0.99 ±0.00 0.67 ±0.29	0.97 ±0.02 0.54 ±0.29	0.92 ±0.06 0.45 ±0.26	0.89 ±0.07 0.36 ±0.21	0.84 ±0.10 0.31 ±0.19
	n = 200 n = 200	R-FAQ R-LWCD R-FAQ R-LWCD	1.00 ±0.00 0.99 ±0.00 0.75 ±0.23 0.74 ±0.22	1.00 ±0.00 1.00 ±0.00 0.62 ±0.26 0.62 ±0.25	0.99 ±0.04 0.99 ±0.04 0.53 ±0.28 0.51 ±0.27	0.95 ±0.10 0.94 ±0.09 0.45 ±0.27 0.44 ±0.25	0.94 ±0.10 0.94 ±0.10 0.38 ±0.23 0.37 ±0.23	0.90 ±0.15 0.90 ±0.16 0.33 ±0.23 0.33 ±0.21
		spectral spectral	0.98 ±0.00 0.96 ±0.03	0.98 ±0.00 0.93 ±0.05	0.96 ±0.00 0.86 ±0.11	0.95 ±0.01 0.76 ±0.18	0.94 ±0.01 0.71 ±0.19	0.93 ±0.01 0.67 ±0.21
		GnCR GnCR	0.99 ±0.00 0.90 ±0.21	0.99 ±0.00 0.80 ±0.28	0.98 ±0.00 0.71 ±0.31	0.97 ±0.00 0.60 ±0.31	0.96 ±0.00 0.62 ±0.29	0.95 ±0.05 0.55 ±0.31
	n = 500 n = 500	FAQ LWC D FAQ LWC D	0 . 9 9 ±0.00 0 . 9 9 ±0.00 0 . 9 8 ±0.03 0 . 9 8 ±0.03	0.99 ±0.00 0.99 ±0.00 0.95 ±0.06 0.95 ±0.06	0.98 ±0.00 0.98 ±0.00 0.87 ±0.13 0.87 ±0.13	0.97 ±0.00 0.97 ±0.00 0.76 ±0.21 0.76 ±0.21	0.96 ±0.00 0.96 ±0.00 0.72 ±0.22 0.72 ±0.22	0.95 ±0.00 0.95 ±0.00 0.67 ±0.24 0.67 ±0.24
		UBI UBI	0.99 ±0.00 0.97 ±0.02	0.98 ±0.00 0.95 ±0.04	0.97 ±0.00 0.88 ±0.14	0.96 ±0.01 0.76 ±0.24	0.95 ±0.00 0.71 ±0.25	0.94 ±0.00 0.67 ±0.25
		Manopt Manopt	0.99 ±0.00 0.97 ±0.03	0.98 ±0.00 0.94 ±0.06	0.97 ±0.00 0.86 ±0.12	0.95 ±0.00 0.76 ±0.18	0.94 ±0.01 0.72 ±0.19	0.93 ±0.01 0.67 ±0.22
		⌘-Spectr. ⌘-Spectr.	1.00 ±0.00 1.00 ±0.00	1.00 ±0.00 1.00 ±0.00	0.99 ±0.00 0.99 ±0.00	0.99 ±0.00 0.96 ±0.12	0.99 ±0.00 0.88 ±0.18	0.98 ±0.00 0.81 ±0.24
		HGnCR HGnCR	1.00 ±0.00 1.00 ±0.00	1.00 ±0.00 0.96 ±0.18	0.99 ±0.00 0.87 ±0.28	0.99 ±0.00 0.80 ±0.32	0.99 ±0.00 0.70 ±0.36	0.99 ±0.00 0.75 ±0.33
	n = 500 n = 500	H-FAQ H-LWCD H-FAQ H-LWCD	1.00 ±0.00 1.00 ±0.00 0.99 ±0.02 0.99 ±0.03	1.00 ±0.00 1.00 ±0.00 0.98 ±0.06 0.97 ±0.07	1.00 ±0.00 1.00 ±0.00 0.91 ±0.13 0.90 ±0.13	0.99 ±0.00 0.99 ±0.00 0.82 ±0.21 0.80 ±0.21	0.99 ±0.00 0.99 ±0.00 0.78 ±0.23 0.77 ±0.23	0.99 ±0.00 0.99 ±0.00 0.74 ±0.26 0.72 ±0.25
		H-UBI H-UBI	1.00 ±0.00 1.00 ±0.00	1.00 ±0.00 1.00 ±0.00	0.99 ±0.00 0.99 ±0.00	0.99 ±0.00 0.98 ±0.07	0.99 ±0.00 0.95 ±0.13	0.98 ±0.00 0.92 ±0.17
		H-Manopt H-Manopt	1.00 ±0.00 1.00 ±0.00	1.00 ±0.00 0.99 ±0.01	1.00 ±0.00 0.93 ±0.12	0.99 ±0.00 0.81 ±0.21	0.98 ±0.01 0.76 ±0.22	0.97 ±0.01 0.72 ±0.25
	n = 500 n = 500	R-FAQ R-LWCD R-FAQ R-LWCD	1.00 ±0.00 1.00 ±0.00 0.99 ±0.03 0.98 ±0.03	1.00 ±0.00 1.00 ±0.00 0.97 ±0.07 0.96 ±0.06	1.00 ±0.00 1.00 ±0.00 0.90 ±0.13 0.89 ±0.13	1.00 ±0.00 1.00 ±0.00 0.80 ±0.21 0.80 ±0.21	1.00 ±0.00 1.00 ±0.00 0.76 ±0.23 0.76 ±0.23	1.00 ±0.00 1.00 ±0.01 0.72 ±0.25 0.71 ±0.25

Table D .

 D 1: Results for Seriation with Duplications on dense, strong-R matrices (with several values of the parameter and N/n), and no noise added. Table D.2: Results for Seriation with Duplications on dense, strong-R matrices (with several values of the parameter and N/n), and noiseProp=5%.Table D.6: Results of synthetic experiments for Seriation+Clustering with Duplications from dense, strong-R matrices of size n = 200, with an additive block matrix with 10 clusters, with Algorithm 5.1 (that do not take the cluster structure into account), denoted SerDupli and Algorithm 5.3, denoted SerDuClus. Both are used with the ⌘-Spectral method at step 3 of the alternate projections Algorithm. The results are averaged over 20 experiments and the standard deviation is given after the ± sign. ±2.199e+06 30.6 ±15.1 26.6 ±13.0 SerDuClus 6.618e+06 ±1.317e+06 13.2 ±13.5 12.7 ±11.3 Résumé Les technologies de séquençage d'ADN ne permettent de lire que de courts fragments, dont on ignore la position sur le génome. L'assemblage de novo vise à reconstituer une séquence d'ADN entière en mettant ces fragments bout-à-bout, tel un puzzle. Dans l'approche OLC (overlap-layout-consensus), on calcule le chevauchement entre fragments afin de les disposer en ordre (réarrangement), puis extraire une séquence consensus. Le réarrangement peut s'écrire comme un problème combinatoire de sériation, où l'on réordonne des éléments comparable entre eux, de sorte que deux éléments adjacents sont similaires. Ce problème est résolu efficacement par un algorithme spectral en l'absence de bruit, mais il en va autrement des données génomiques réelles. En particulier, des régions du génome sont similaires bien qu'éloignées (séquences répétées), rendant l'assemblage problématique. Les méthodes d'assemblage emploient des algorithmes hiérarchiques et gloutons pour désambiguïser les séquences répétées. Nous proposons ici une approche épurée où l'on réarrange tous les fragments « d'un coup »via la résolution de sériation. Notre première contribution montre que l'emploi de la méthode spectrale pour le réarrangement s'intègre parfaitement dans le schéma OLC, produisant des résultats de qualité semblable aux méthodes standard. Cependant, du fait des séquences répétées, cette méthode produit des assemblages fragmentés (typiquement en quelques sous-séquences au lieu d'une). La deuxième contribution est un prolongement de la méthode spectrale lié à la réduction de dimension sous conservation de distances, englobant les problèmes de sériation et de sériation circulaire (une variante où les éléments peuvent être ordonnés selon un cycle) dans un cadre unifié. Ce prolongement rend l'algorithme robuste au bruit et résout le problème de fragmentation de l'assemblage précédent. Notre troisième contribution formalise la sériation robuste, où l'on souhaite réordonner des données bruitées. Nous décrivons des liens avec d'autres problèmes combinatoires, en particulier pour des matrices modélisant les données réelles d'ADN. Nous proposons des algorithmes adaptés, améliorant expérimentalement la robustesse sur données synthétiques et réelles, bien que moins clairement que la deuxième contribution.

		N/n N/n	method method	d2S d2S	Huber (x1e-7) meanDist stdDist Time (x1e-2s) Huber (x1e-7) meanDist stdDist Time (x1e-2s)
		1.33 1.33	spectral 0.03 ±0.00 spectral 0.07 ±0.00 ⌘-Spectr. 0.03 ±0.00 ⌘-Spectr. 0.07 ±0.00 H-UBI 0.02 ±0.00 H-UBI 0.07 ±0.00	8.33 ±0.01 8.36 ±0.01 8.33 ±0.01 8.36 ±0.01 8.33 ±0.01 8.35 ±0.02	3.0 ±0.7 5.7 ±0.9 3.0 ±0.7 5.7 ±0.9 2.8 ±0.7 5.2 ±0.9	5.5 ±1.0 7.2 ±1.3 5.5 ±1.0 7.2 ±1.2 5.2 ±1.0 6.4 ±1.4	1.27 ±0.78 5.14 ±1.36 1.39 ±0.80 5.75 ±1.41 6.37 ±1.60 1.48 ±0.93
	0.1 0.1	2 2	spectral 0.03 ±0.00 ⌘-Spectr. 0.03 ±0.00 spectral 0.07 ±0.00 ⌘-Spectr. 0.07 ±0.00 H-UBI 0.03 ±0.00 H-UBI 0.07 ±0.00	8.37 ±0.01 8.37 ±0.01 8.38 ±0.01 8.38 ±0.01 8.37 ±0.01 8.37 ±0.01	7.1 ±1.0 7.1 ±1.0 8.5 ±0.8 8.5 ±0.8 7.0 ±1.0 8.4 ±0.8	7.5 ±1.0 7.6 ±1.0 7.7 ±0.8 7.7 ±0.9 7.5 ±0.9 7.5 ±0.9	5.05 ±1.05 6.62 ±4.69 7.62 ±5.05 5.41 ±1.07 6.14 ±1.17 8.75 ±6.02
		4 4	spectral 0.02 ±0.00 spectral 0.06 ±0.00 ⌘-Spectr. 0.02 ±0.00 ⌘-Spectr. 0.06 ±0.00 H-UBI 0.03 ±0.00 H-UBI 0.06 ±0.00	8.35 ±0.01 8.35 ±0.01 8.35 ±0.01 8.35 ±0.01 8.35 ±0.01 8.35 ±0.01	12.8 ±2.2 13.7 ±2.4 12.9 ±2.3 13.8 ±2.3 13.1 ±1.4 13.8 ±2.2	7.8 ±1.8 7.9 ±2.7 7.9 ±2.1 8.0 ±2.7 7.9 ±1.4 7.9 ±2.7	5.15 ±1.49 5.41 ±2.03 5.47 ±1.58 5.86 ±2.16 7.05 ±2.36 6.17 ±1.58
		1.33 1.33 N/n	spectral 0.25 ±0.04 spectral 0.27 ±0.04 ⌘-Spectr. 0.15 ±0.02 ⌘-Spectr. 0.17 ±0.02 H-UBI 0.24 ±0.04 H-UBI 0.25 ±0.03 method	1.36 ±0.03 1.37 ±0.03 1.30 ±0.01 1.31 ±0.01 1.35 ±0.03 1.36 ±0.02 Huber	6.1 ±1.8 6.7 ±1.8 2.2 ±0.7 2.6 ±0.7 5.5 ±1.6 5.6 ±1.5 meanDist stdDist 7.9 ±1.6 8.4 ±1.6 3.7 ±1.1 4.1 ±1.0 7.3 ±1.4 7.3 ±1.4	1.60 ±0.58 8.74 ±4.85 1.61 ±0.78 6.12 ±4.84 11.06 ±7.56 2.01 ±0.74
	0.5 0.5	2 4 2 4 1.33 1.33 8.086e+06 2 spectral 0.27 ±0.02 1.41 ±0.02 ⌘-Spectr. 0.22 ±0.02 1.37 ±0.02 H-UBI 0.26 ±0.02 1.40 ±0.02 spectral 0.18 ±0.01 1.35 ±0.01 ⌘-Spectr. 0.18 ±0.01 1.35 ±0.01 spectral 0.28 ±0.02 1.41 ±0.02 ⌘-Spectr. 0.23 ±0.02 1.37 ±0.02 H-UBI 0.26 ±0.02 1.40 ±0.02 spectral 0.19 ±0.01 1.35 ±0.01 ⌘-Spectr. 0.19 ±0.01 1.35 ±0.01 SerDupli SerDupli 1.111e+07 ±2.599e+06 SerDuClus 9.271e+06 ±2.341e+06 H-UBI 0.19 ±0.01 1.35 ±0.01 spectral 0.61 ±0.02 2.10 ±0.13 ⌘-Spectr. 0.30 ±0.06 1.48 ±0.08 H-UBI 0.30 ±0.12 1.50 ±0.15 H-UBI 0.32 ±0.11 1.50 ±0.16 1.33 ⌘-Spectr. 0.32 ±0.06 1.49 ±0.08 H-UBI 0.19 ±0.01 1.35 ±0.01 spectral 0.62 ±0.02 2.10 ±0.13 SerDupli 2.125e+07 ±2.843e+06 42.8 ±10.9 9.5 ±1.6 6.6 ±1.5 9.0 ±1.5 14.4 ±2.8 14.3 ±2.9 9.7 ±1.5 6.7 ±1.4 9.0 ±1.5 14.4 ±2.4 14.2 ±2.8 40.0 ±8.3 28.4 ±9.7 14.8 ±2.5 15.2 ±2.4 2.2 ±1.4 2.4 ±2.1 2.6 ±2.3 2.3 ±1.1 14.8 ±2.6 15.3 ±2.4 4 SerDuClus 1.504e+07 ±2.835e+06 35.0 ±8.7	8.4 ±1.3 6.7 ±1.9 8.1 ±1.2 8.7 ±2.7 8.9 ±2.9 8.5 ±1.2 6.6 ±1.9 8.1 ±1.3 8.4 ±2.1 8.7 ±2.6 27.4 ±6.8 20.2 ±7.4 8.8 ±2.1 15.2 ±1.4 3.1 ±1.5 3.1 ±2.1 3.2 ±2.3 3.2 ±1.3 8.8 ±2.2 15.3 ±1.3 19.6 ±5.1 17.0 ±4.2	7.47 ±3.20 7.89 ±3.89 10.09 ±4.90 6.53 ±1.90 1.07 ±0.58 1.08 ±0.64 1.46 ±0.91 6.21 ±1.85 6.72 ±1.72 7.59 ±2.28 8.62 ±2.46 26.12 ±2.96 26.30 ±3.52 15.35 ±7.54 18.45 ±6.40 9.04 ±8.61 7.86 ±1.89 9.20 ±8.34
	1 1	2 2	spectral 0.60 ±0.03 ⌘-Spectr. 0.42 ±0.04 spectral 0.61 ±0.03 ⌘-Spectr. 0.42 ±0.04 H-UBI 0.49 ±0.05 H-UBI 0.49 ±0.05	2.46 ±0.11 1.91 ±0.13 2.46 ±0.12 1.92 ±0.13 2.06 ±0.14 2.06 ±0.15	19.3 ±6.6 10.3 ±8.6 19.4 ±6.7 10.6 ±8.8 10.4 ±7.9 10.3 ±7.9	12.6 ±4.9 9.8 ±6.4 12.6 ±4.9 10.1 ±6.6 8.5 ±6.0 8.4 ±6.1	1.78 ±0.31 2.13 ±0.66 1.52 ±0.75 1.20 ±0.51 2.57 ±0.21 3.43 ±0.80
		4 4	spectral 0.37 ±0.02 spectral 0.37 ±0.02 ⌘-Spectr. 0.34 ±0.01 ⌘-Spectr. 0.35 ±0.01 H-UBI 0.36 ±0.01 H-UBI 0.37 ±0.02	1.81 ±0.05 1.80 ±0.05 1.78 ±0.04 1.79 ±0.04 1.80 ±0.04 1.80 ±0.05	19.3 ±4.7 19.0 ±4.9 20.0 ±6.9 20.0 ±6.8 18.9 ±5.2 18.9 ±4.9	11.6 ±4.4 11.2 ±4.6 13.2 ±6.1 13.2 ±6.0 11.5 ±4.8 11.3 ±4.4	1.44 ±0.30 1.96 ±0.50 0.77 ±0.17 1.00 ±0.34 2.25 ±0.65 1.83 ±0.41

(a) Linear Banded (b) Circular Banded

Remerciements

circle have high similarity. Thus, while the repeats make it impossible to find a linear ordering consistent with all pairwise similarity information, we can hope that the chain structure appears in a higher dimensional embedding, where the repeats may cause angles and loops in the chain.

A significant part of this chapter is devoted to existing work, since several results scattered in different fields (from theoretical to application-specific) provide intuition or partial results motivating our approach. Bringing them all together into a consistent frame is one of the contributions of this work.

A remark about the notations used in this Chapter. We have previously used the notation 1 (L) ... n (L) to denote the eigenvalues of the (laplacian) matrix L of size n. Here, we will instead use 0 (L) ... n 1 (L), and the same indexing for the associated eigenvectors, as we will be interested only in the non-zero eigenvalues and the associated eigenvectors, hence we start the indexing to 1 from the second (which is the first non-zero) eigenvalue.

The content of this chapter is based on the following publication, Antoine Recanati, Thomas Kerdreux, and Alexandre d'Aspremont. Reconstructing latent orderings by spectral clustering. arXiv preprint arXiv:1807.07122,2 0 1 8 a .

Supplementary for this chapter is given in Appendix Chapter B.

Chapter Abstract

Spectral clustering uses a graph Laplacian spectral embedding to enhance the cluster structure of some data sets. When the embedding is one dimensional, it can be used to sort the items (spectral ordering). Empirically we found that a multidimensional Laplacian embedding enhances the latent ordering of the data, if any. This also extends to circular orderings, a case where unidimensional embeddings fail. We tackle the task of retrieving linear and circular orderings in a unifying framework, and show how a latent ordering on the data translates into a filamentary structure on the Laplacian embedding. We propose a method to recover it, illustrated with numerical experiments on synthetic data, real DNA third-generation sequencing data, and spatial conformation Hi-C data. The code and experiments are available at https://github.com/antrec/mdso.

Chapter 4

Robust Seriation

The work presented so far in this manuscript relies mainly on the spectral relaxation of the 2-SUM problem, introduced in Chapter 1. In Chapter 2, we use the existing method as is in the context of de novo genome assembly. In Chapter 3, we propose an enhancement of the spectral method, leading to a more efficient approach to overcome the presence of repeat-induced noise in the similarity matrices. Here, we explore another strategy, following the convex relaxations approaches to permutation problems proposed by, e.g., Vogelstein et al. [2011], Fogel et al.

[2013], Lim and Wright [2014,2016], Evangelopoulos et al. [2017a]. We aim to model the similarity matrices arising in de novo assembly, and design algorithmic schemes that are robust to the specific, repeat-induced noise.

The content of this chapter is based on the following publication, Antoine Recanati, Nicolas Servant, Jean-Philippe Vert, and Alexandre d'Aspremont. Robust seriation and applications to cancer genomics. arXiv preprint arXiv:1806.00664,2 0 1 8 b Supplementary material for this chapter is given in Appendix Chapter C.

Chapter Abstract

The seriation problem seeks to reorder a set of elements given pairwise similarity information, so that elements with higher similarity are closer in the resulting sequence. When a global ordering consistent with the similarity information exists, an exact spectral solution recovers it in the noiseless case and seriation is equivalent to the combinatorial 2-SUM problem over permutations, for which several relaxations have been derived. However, in applications such as DNA assembly, similarity values are often heavily corrupted, and the solution of 2-SUM may no longer yield an approximate serial structure on the elements. We introduce the robust seriation problem and show that it is equivalent to a modified 2-SUM problem for a class of similarity matrices modeling those observed in DNA assembly. We explore several relaxations of this modified 2-SUM problem and compare them empirically on both synthetic matrices and real DNA data. algorithm to (approximately) solve (2-SUM)i nl i n e3. Here, is a parameter that controls the influence of the previous iterates of ⌘, the case =0is just plain alternate minimization.

In practice, we evaluate the objective of (HuberSUM( )) for A and ⇡ (t) at each iteration, and keep the iterate ⇡ with the lowest score.

First Order Optimization on Manifold

Finally, we used a manifold optimization toolbox [START_REF] Boumal | Manopt, a Matlab toolbox for optimization on manifolds[END_REF]a sab l a c k -b o x ,t o which we provide the expression of the objective and gradient of HuberSUM( ) and ask for the minimum over the sphere (computed with a trust-regions algorithm). We refer to this method as Manopt, which is the name of the toolbox [START_REF] Boumal | Manopt, a Matlab toolbox for optimization on manifolds[END_REF]. We use the formulation of the hyperplane H n through an affine transformation as with the UBI method, in order to use (Manopt) with f Hn on a sphere in R n 1 in the experiments.

Numerical Results

In this section, we test the algorithms detailed above on both synthetic and real data sets.

Synthetic data

We performed experiments with matrices from M n ( ,s) with n = 100, 200, 500, = n/10,n/20, and s/s lim =0.5, 1, 2.5, 5, 7.5, 10, with s is the number of out-of-band terms as in Definition 4.2.1 and s lim =(n 1) is the value appearing in Proposition 4.2.2, where R2SUM( ) and Robust Seriation coincide when s  s lim .I nT a b l e4.1, we show the seriation results of the different methods described in Section 4.3. When an algorithm can be used for 2-SUM, but also with R2SUM( ) (or HuberSUM( ), respectively), we pre-pend -R (or -H, resp.) to its name in the R-2SUM (or Huber, resp.) corresponding row of the Table . In Table 4.2, we show the Kendall-⌧ score for different values of s/s lim . For a given set of parameters (n, ,s), we generated 100 experiments with random locations for the out-of-band entries. The results displayed in Uncons is also competitive. Some methods such as HGnCR do not perform as good in average, but have a higher standard deviation over the 100 simulations. They actually perform well on most simulations, but fail on a few ones. Overall this results in a lower mean Kendall-⌧ score and a higher standard deviation.

Chapter 5

Seriation with Duplications

In this chapter, we introduce the problem of Seriation with Duplications. It is an extension of Seriation that differs from the problem of seriation with repeats. Here, given two duplicates, we do not observe the similarity values for any of the two duplicates, like we could with two repeated reads in genome assembly. Instead, we observe an aggregated similarity over all duplicates. It is motivated by an application to cancer genome assembly given Hi-C frequency data.

The content of this chapter is based on the following publication, Antoine Recanati, Nicolas Servant, Jean-Philippe Vert, and Alexandre d'Aspremont. Robust seriation and applications to cancer genomics. arXiv preprint arXiv:1806.00664,2 0 1 8 b Supplementary material for this chapter is given in Appendix Chapter D.

Chapter Abstract

The seriation problem seeks to reorder a set of elements given pairwise similarity information, so that elements with higher similarity are closer in the resulting sequence. We introduce the problem of seriation with duplications, which is a generalization of Seriation motivated by applications to cancer genome reconstruction.

In this context, we also aim to reorder a set of elements such that similar elements are nearby. However, some of these elements are identical copies, but we do not have access to the similarity information involving each of the copies. Instead, we observe a coarser-grained aggregated similarity, which is the sum over all the copies.

We propose an alternated minimization scheme that involves seriation, and present preliminary results on synthetic data sets.

with Duplication setting (SD), the matrices may not fit the class M n ( ,s), especially when the matrix S to be recovered is dense (and not a band matrix). Second, the output of the algorithm is a permutation Π ⇤ , but what we are really interested in step 3 of Algorithm 5.1 is the matrix S ⇤ 2 R ⇤ N that is the closest to S (t) .T oa p p r o x i m a t eS ⇤ 2 R ⇤ N ,w efi r s tu s eo n eo f the methods introduced in Section 4.2 to find a permutation Π ⇤ that makes Π ⇤ S (t) Π T ⇤ as close to R ⇤ N as possible. Still, in general the permuted matrix Π ⇤ S (t) Π T ⇤ will not be in R ⇤ N . We then project Π ⇤ S (t) Π T ⇤ onto R ⇤ N , which is solved with linear programming. Indeed, the projection, in `1 norm for example of a matrix S, reads minimize

We can also use a Froebenius norm and consider the sum of squares instead of the absolute differences. We would then use quadratic programming, as we have then a quadratic objective with linear constraints. The constraint R 2 R ⇤ N can indeed be written as linear constaints on R. Specifically, we consider the vectorized forms of S and R, s, r 2 R N 2 , which are the concatenation of the columns of S and R, respectively. Imposing R 2 R ⇤ is equivalent to saying that r u  r v for all pairs of indexes (u, v) such that the corresponding subscripts for u are on a diagonal higher than those for v. There is one linear constraint per pair (u, v) (and there are N (N 1) 2 pairs), but we can reduce the number of constraints by adding slack variables { k } 1kN and impose that for each element r u on a given diagonal k, 1  k  N 1, r u  k+1 and r u k . Finally, we can use aprioriknowledge on how the values are supposed to decrease when moving away from the diagonal (e.g., a power law S ij = |i j| as in our experiments, which is consistent with the intra-chromosomal frequency observed in Lieberman-Aiden et al. In step 4 of Algorithm 5.1, we wish to compute the projection of S on the set of matrices X that satisfy ZXZ T = A, that is to say, solve the following optimization problem on variable Appendix A Supplementary Material for Chapter 2, Application of the Spectral Method to Genome Assembly

A.1 Running Times

In Table A.1, we give the running time of the methods evaluated in Chapter 2.F i g u r eA.1 focuses on the runtime for the layout method (spectral algorithm) only.

A key ap o s t e r i o r iremark is that the implementation of the spectral method whose results are reported here is quite slow for large matrices, as one can see in Figure A.1. The hack proposed here was to use the Julia computing language [Bezanson et al., 2017] for large matrices.

However, we noted during the experiments of Chapter 3 that a simpler solution could be used, while keeping all the code in python. Indeed, resorting to the pyamg solver instead of arpack in the eigenvalue computation of the Laplacian solved the issue and enabled a speedup of an order of magnitude for matrices with n ⇠ 10 4 .

A.1.1 Total time

Table A.1 shows the run-time and peak memory for the previously compared methods, when run on a 24 cores Intel Xeon E5-2640 2.50GHz node. Runtime and Max mem correspond to the wall-clock and maximum resident set size fields of the unix /usr/bin/time -v command. The first column (Spectral Layout) displays the running time of the layout phase of our method in the following way: time to reorder contigs with the spectral algorithm (total time to get fine-grained layout); the total time for the layout (including the fine-grained computation of the position of the reads on a backbone sequence) is given between parentheses next to the time for the ordering. The second column gives the runtime for our full pipeline, including running minimap to obtain the overlaps. The runtime for Racon includes the time to map the reads than 3000 in the code since its eigenvector computation scales better for large matrices but has a non-negligible overhead for small matrices. However, as mentioned earlier, a simpler and more efficient solution is to switch to the amg solver instead of arpack in the eigenvalue computation (results not shown here).

A.2 the Bandwidth Heuristic

We present some qualitative and quantitative results to support the bandwidth heuristic. dataset with option -perfect, which means these synthetic reads follow the same length distribution than the original dataset, but have no errors, and have the coverage specified above.

The matrices A.3b and A.3e were generated from the A. baylyi ONT R7.3 dataset without the -perfect option, which means they have the same length and error distribution than the original data, but with higher coverage.

For perfect and noisy synthetic A. baylyi reads and with sufficient coverage, all outliers could be removed by thresholding while keeping a connected similarity graph (all matrices in the Figure are connected). On the other hand, the similarity matrix generated with S. cerevisiae perfect reads still harbors a few outliers after removing 90% of the overlaps (with lowest score).

When increasing the threshold value, the connectivity within some individual chromosomes will be broken before all outliers have been removed. Additional structural information (as used in Canu or Miniasm) will be required to resolve repeats in such situations. ). However, this heuristic is not applicable when the size of the similarity matrix is small. For instance, if the matrix is of size 100, the bandwidth cannot exceed 100 and the use of the heuristic is precluded. • QUASThttps://sourceforge.net/projects/quast/files/

• GraphMaphttps://github.com/isovic/GraphMap

• errorrates.py from samscriptshttps://github.com/isovic/samscripts

• NanoSimhttps://github.com/bcgsc/NanoSim SPOA is used in our pipeline for performing multiple sequence alignment. For generating the consensus in windows, it was run with the options : -l 2 -r 0 -x -3 -o -5 -e -2

(semi-global alignment with custom gap and mismatch penalties). minimap was run with options -Sw5 -L100 -m0 -t12 (long reads specific values and multithreading with 12 threads).

miniasm was run with default parameters when used as a comparative method. Canu was run with saveReadCorrections=True option and data specifications (e.g., genomeSize=3.6m

-nanopore-raw). Racon was run with the alignment generated with minimap (to map the draft assembly, either from miniasm or from our pipeline) with default parameters. GraphMap [START_REF] Vaser | Racon-rapid consensus module for raw de novo genome assembly of long uncorrected reads[END_REF] was used to generate alignment between the reads and the reference genome in order to have the position of the reads and their error rate (which was computed with the script errorrates.py). DNAdiff and QUAST were used to evaluate the assemblies. To concatenate the contigs obtained with our method, we extracted their ends (end length used : 35kbp) and used minimap with options -Sw5 -L500 to compute overlaps between them, and ran miniasm with options -1 -2 -e 0 -c 0 -r 1,0 (no pre-selection, no cutting small unitigs, no overlap drop). The related script is available in the tools folder of our GitHub code. We also publish the other scripts we used (although they may be poorly written and undocumented), including our implementation of the optical mapping algorithm of Nagarajan et al. [2008], in the tools folder. Notation: We will commonly denote a permutation of {1,...,n} and S the set of all such permutations. When represented matricially, will often be noted Π while cyclic permutation of {1,...,n} will be noted as ⌧ . A will usually denote the matrix of raw pair-wise similarities. S will denote the similarity matrix resulting from Algorithm 3.3,a n dk a neighboring parameter.

Finally we use indexed version ⌫ (resp., ) to denote eigenvalues of a similarity matrix (resp.

a graph Laplacian).

B.1 Additional Algorithms

B.1.1 Merging connected components

The new similarity matrix S computed in Algorithm 3.3 is not necessarily the adjacency matrix of a connected graph, even when the input matrix A is. For instance, when the number of nearest neighbors k is low and the points in the embedding are non uniformly sampled along a curve, S may have several, disjoint connected components (let us say there are C of them in the following). Still, the baseline Algorithm 3.1 requires a connected similarity matrix as input. When S is disconnected, we run 3.1 separately in each of the C components, yielding C sub-orderings instead of a global ordering.

However, since A is connected, we can use the edges of A between the connected components to merge the sub-orderings together. Specifically, given the C ordered subsequences, we build a meta similarity matrix between them as follows. For each pair of ordered subsequences (c i ,c j ),

On the difficult instances, the baseline and our method perform more poorly than on easy instances. However, we also computed the gain over the baseline, i.e.,t h ed i fference of score between our method and the baseline, for each experiment, and it is always, or almost always positive, i.e., our method almost always beats the baseline although the confidence intervals cross each other.

B.2.3 Numerical results with KMS matrices

In We observe that the method performs roughly equally well with k in a range from 5 to 20, and that the performances drop when k gets too large, around k = 30. This can be interpreted as follows. When k is too large, the assumption that the points in the embedding are locally fitted by a line no longer holds. Note also that in practice, for small values of k, e.g., k =5,the new similarity matrix S can be disconnected, and we have to resort to the merging procedure described in Algorithm B.1.

B.2.5 Sensitivity to the normalization of the Laplacian

We performed experiments to compare the performances of the method with the default Lapla- We interpret these results as follows. With the (d-LE), Algorithm 3.3,l i n e5 treats equally all dimensions of the embedding. However, the curvature of the embedding tends to increase with the dimension (for C R matrix, the period of the cosines increases linearly with the dimension).

The filamentary structure is less smooth and hence more sensitive to noise in high dimensions, which is why the results are improved by damping the high dimensions (or using a reasonably small value for d).

Using cos (⇡/2 x)=sin(x) and sin (⇡/2 x) = cos (x), we thus have,

To conclude, let us observe that f (q) is non-increasing on [1,p/2]. Informally, the terms {z 1 k } 1kq appearing in the partial sums S

q are all non-negative for q  p/2. Formally, remark that the derivative of f , df /dq(q)=( ⇡/n) cos ⇡(2q + 1)/n is non-negative for q 2 [1,p/2].

Hence, for q  p/2, f (q 1/2) f (q), which ends the proof of equation (B.3).

To get the first equality of (B.5), from the exact form in (B.7), we have (n =2p)

For the second equality in (B.5), we have (m =2q):

Finally to get (B.6), let us write (n =2p and m odd):

The following lemma gives an important property of the partial sum of the z = cos(2⇡km/n). Consider first n =2 p and m even. For m =1,...,p and q =1,...,p 2

Otherwise we have for every (m, q) 2 {1,...,p} 2

S (1) q >S (m) q , (B.10) with equality when q = p.

Proof. Case m and n coprime. Values of z (m) k k=1,...,p are all distinct. Indeed z

implies that n divides k + k 0 or k k 0 .I ti si m p o s s i b l e( t h er a n g eo fk

Case m and n not coprime. m = dm 0 and n = dn 0 , with d 3. In that situation we need to distinguish according to the parity of n.

Case n =2 p +1. Let's first remark that z (1) k k=1,...,p takes all values but two ( 1 and 1) of the cosinus of multiple of the angle 2⇡ n , e.g. z

(1) k k=1,...,p ⇢ Z n .A l s o (z

k ) k=1,...,p is non-increasing.

Let's prove (B.10) by distinguishing between the various values of q.

• Consider q = p (n 0 1),...,p.F r o m( B.2) in lemma (B.3.2), we have S • For q =1 ,...,n 0 1 it is the same type of argument. Indeed the (z

k ) k takes the highest values in Z n in decreasing order, while (z

). This concludes (B.10).

Note that when n 0 p+1 2 ,( B.10) is then true for all q. In the sequel, let's then assume that this is not the case, e.g. n 0 < p+1 2 .

• For q = n 0 1,...,

q are non-negative. Hence S

q is non-decreasing and lower bounded by S (1)

for k =1 ,...,n 0 ,i ti st r u e that for all q in the considered set, S (m) q is upper-bounded by S

(1)

. All in all it shows (B.10) for these values of q.

• For q = ⌅ p 2 ⇧ +1,...,p n 0 ,w ea p p l y( B.3) with q = n 0 (and indeed

q ) is non-increasing for the considered sub-sequence of q,( B.10) is true.

Case n =2 p. Here z (1) k k=1,...,p takes unique values in Z n [ { 1}. We also need to distinguish according to the parity of m.

• z (m) k k=1,...,n 0 1 takes also unique value in Z n . We similarly get (B.10) for q =1,...,n 0 1, and for q = n 0 because S (m)

p = 1 so that we can do the same reasoning as with n odd to prove (B.10) for q = p n 0 +1,...,p and q =1 ,...,n 0 . The remaining follows from the symmetry property (B.4) of the sequence (S (1) q ) q in Lemma B.3.1.

• m and n even, we have that S

(1)

for q<p 1 follows with same techniques as before.

B.3.2 Properties on R-Toeplitz circular matrix.

This proposition is a technical method that will be helpful at proving that the eigenvalues of a R-circular Toeplitz matrix are such that ⌫ 1 > ⌫ m .

Proposition B.3.3. Suppose than for any k =1,...,q :

with (w i ) and (w i ) two sequences of reals. Then, if (b k ) k is non increasing and non negative, we have

(B.12)

Proof. We have

As soon as there exists k 0 2 {1,...,q} such that

Consider n and m even. We now need to prove that

k and wk = z

The last inequality results from the monotonicity of (b k ) and is equivalent to (B.22). It concludes the proof.

B.3.3 Recovering exactly the order.

Here we provide the proof for Theorem 3.3.2.

Theorem B.3.6. Consider the seriation problem from an observed matrix ΠSΠ T ,wher eS is a Rcircular Toeplitz matrix. Denote by L the associated graph Laplacian. Then the two dimensional laplacian spectral embedding ( (d-2SUM) with d=2) of the items lies ordered and equally spaced on a circle.

T h e eigenspace associated to its second smallest eigenvalue corresponds to that of µ 1 in A. A and S share the same spectrum. Hence the eigenspace of µ 1 in A is composed of the two vectors Πy 1,sin and Πy 5.2) on dense matrices expanding the results from Section 5.4. Tables D.4

and D.5 expand these results to matrices in M N ( ,s).