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Anomaly detection in multivariate signals is a task of paramount importance in many disciplines (epidemiology, finance, cognitive sciences and neurosciences, oncology, etc.). In this perspective, Topological Data Analysis (TDA) offers a battery of "shape" invariants that can be exploited for the implementation of an effective detection scheme. Our contribution consists of extending the constructions presented in \cite{chretienleveraging} on the construction of simplicial complexes from the Signatures of signals and their predictive capacities, rather than the use of a generic distance as in \cite{petri2014homological}. Signature theory is a new theme in Machine Learning arXiv:1603.03788 stemming from recent work on the notions of Rough Paths developed by Terry Lyons and his team \cite{lyons2002system} based on the formalism introduced by Chen \cite{chen1957integration}. We explore in particular the detection of changes in topology, based on tracking the evolution of homological persistence and the Betti numbers associated with the complex introduced in \cite{chretienleveraging}. We apply our tools for the analysis of brain signals such as EEG to detect precursor phenomena to epileptic seizures., Comment: 14 pages, 5 figures Under review for Journ\'ee des Statistiques 2024

Hardware implementation of neural network are an essential step to implement next generation efficient and powerful artificial intelligence solutions. Besides the realization of a parallel, efficient and scalable hardware architecture, the optimization of the system's extremely large parameter space with sampling-efficient approaches is essential. Here, we analytically derive the scaling laws for highly efficient Coordinate Descent applied to optimizing the readout layer of a random recurrently connection neural network, a reservoir. We demonstrate that the convergence is exponential and scales linear with the network's number of neurons. Our results perfectly reproduce the convergence and scaling of a large-scale photonic reservoir implemented in a proof-of-concept experiment. Our work therefore provides a solid foundation for such optimization in hardware networks, and identifies future directions that are promising for optimizing convergence speed during learning leveraging measures of a neural network's amplitude statistics and the weight update rule., Comment: Submitted to ECML-PKDD workshop Deep Learning meets Neuromorphic Hardware

This work addresses the Robust counterpart of the Shortest Path Problem (RSPP) with a correlated uncertainty set. Since this problem is hard, a heuristic approach, based on Frank-Wolfe's algorithm named Discrete Frank-Wolf (DFW), has recently been proposed. The aim of this paper is to propose a semi-definite programming relaxation for the RSPP that provides a lower bound to validate approaches such as DFW Algorithm. The relaxed problem results from a bidualization that is done {through} a reformulation of the RSPP into a quadratic problem. Then the relaxed problem is solved using a sparse version of Pierra's decomposition in a product space method. This validation method is suitable for large size problems. The numerical experiments show that the gap between the solutions obtained with the relaxed and the heuristic approaches is relatively small.

Recent extensive numerical experiments in high scale machine learning have allowed to uncover a quite counterintuitive phase transition, as a function of the ratio between the sample size and the number of parameters in the model. As the number of parameters $p$ approaches the sample size $n$, the generalisation error increases, but surprisingly, it starts decreasing again past the threshold $p=n$. This phenomenon, brought to the theoretical community attention in \cite{belkin2019reconciling}, has been thoroughly investigated lately, more specifically for simpler models than deep neural networks, such as the linear model when the parameter is taken to be the minimum norm solution to the least-squares problem, firstly in the asymptotic regime when $p$ and $n$ tend to infinity, see e.g. \cite{hastie2019surprises}, and recently in the finite dimensional regime and more specifically for linear models \cite{bartlett2020benign}, \cite{tsigler2020benign}, \cite{lecue2022geometrical}. In the present paper, we propose a finite sample analysis of non-linear models of \textit{ridge} type, where we investigate the \textit{overparametrised regime} of the double descent phenomenon for both the \textit{estimation problem} and the \textit{prediction} problem. Our results provide a precise analysis of the distance of the best estimator from the true parameter as well as a generalisation bound which complements recent works of \cite{bartlett2020benign} and \cite{chinot2020benign}. Our analysis is based on tools closely related to the continuous Newton method \cite{neuberger2007continuous} and a refined quantitative analysis of the performance in prediction of the minimum $\ell_2$-norm solution., Comment: New section on generalisation added

We study the problem of super-resolution, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. It has been shown that exact recovery is possible by minimising the total variation norm of the measure, and a practical way of achieve this is by solving the dual problem. In this paper, we study the stability of solutions with respect to the solutions dual problem, both in the case of exact measurements and in the case of measurements with additive noise. In particular, we establish a relationship between perturbations in the dual variable and perturbations in the primal variable around the optimiser and a similar relationship between perturbations in the dual variable around the optimiser and the magnitude of the additive noise in the measurements. Our analysis is based on a quantitative version of the implicit function theorem., Comment: 35 pages, 5 figures

Many statistical learning problems have recently been shown to be amenable to Semi-Definite Programming (SDP), with community detection and clustering in Gaussian mixture models as the most striking instances [javanmard et al., 2016]. Given the growing range of applications of SDP-based techniques to machine learning problems, and the rapid progress in the design of efficient algorithms for solving SDPs, an intriguing question is to understand how the recent advances from empirical process theory can be put to work in order to provide a precise statistical analysis of SDP estimators. In the present paper, we borrow cutting edge techniques and concepts from the learning theory literature, such as fixed point equations and excess risk curvature arguments, which yield general estimation and prediction results for a wide class of SDP estimators. From this perspective, we revisit some classical results in community detection from [gu\'edon et al.,2016] and [chen et al., 2016], and we obtain statistical guarantees for SDP estimators used in signed clustering, group synchronization and MAXCUT.

A high efficiency hardware integration of neural networks benefits from realizing nonlinearity, network connectivity and learning fully in a physical substrate. Multiple systems have recently implemented some or all of these operations, yet the focus was placed on addressing technological challenges. Fundamental questions regarding learning in hardware neural networks remain largely unexplored. Noise in particular is unavoidable in such architectures, and here we investigate its interaction with a learning algorithm using an opto-electronic recurrent neural network. We find that noise strongly modifies the system's path during convergence, and surprisingly fully decorrelates the final readout weight matrices. This highlights the importance of understanding architecture, noise and learning algorithm as interacting players, and therefore identifies the need for mathematical tools for noisy, analogue system optimization., Comment: 8 pages, 5 figures

We study the problem of super-resolution, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. It has been recently shown that exact recovery is possible by minimising the total variation norm of the measure. An alternative practical approach is to solve its dual. In this paper, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way., Comment: 4 pages double column

This paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisation, in the context of low rank estimation. Moreover, our Burer-Monteiro factors are shown to lie on a Stiefel manifold. We propose a new generalized Bayesian estimator for this problem and prove novel prediction bounds for clustering. We also devise a componentwise Langevin sampler on the Stiefel manifold to compute this estimator., Comment: Accepted at the LOD 2020 Conference -- The Sixth International Conference on Machine Learning, Optimization, and Data Science

Consider $L$ groups of point sources or spike trains, with the $l^{\text{th}}$ group represented by $x_l(t)$. For a function $g:\mathbb{R} \rightarrow \mathbb{R}$, let $g_l(t) = g(t/\mu_l)$ denote a point spread function with scale $\mu_l > 0$, and with $\mu_1 < \cdots < \mu_L$. With $y(t) = \sum_{l=1}^{L} (g_l \star x_l)(t)$, our goal is to recover the source parameters given samples of $y$, or given the Fourier samples of $y$. This problem is a generalization of the usual super-resolution setup wherein $L = 1$; we call this the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples of $y$, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group $1$ to $L$. In particular, the estimation process at stage $1 \leq l \leq L$ involves (i) carefully sampling the tail of the Fourier transform of $y$, (ii) a \emph{deflation} step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii)., Comment: 50 pages, 10 figures, made notational changes and corrected typos after reviewer feedback, to appear in Journal of Fourier Analysis and Applications

The goal of this note is to study the smallest conic singular value of a matrix from a Lagrangian duality viewpoint and provide an efficient method for its computation.

The increase in the world's population and rising standards of living is leading to an ever-increasing number of vehicles on the roads, and with it ever-increasing difficulties in traffic management. This traffic management in transport networks can be clearly optimized by using information and communication technologies referred as Intelligent Transport Systems (ITS). This management problem is usually reformulated as finding the shortest path in a time varying random graph. In this article, an online shortest path computation using stochastic gradient descent is proposed. This routing algorithm for ITS traffic management is based on the online Frank-Wolfe approach. Our improvement enables to find a confidence interval for the shortest path, by using the stochastic gradient algorithm for approximate Bayesian inference.

In Compressed Sensing and high dimensional estimation, signal recovery often relies on sparsity assumptions and estimation is performed via $\ell_1$-penalized least-squares optimization, a.k.a. LASSO. The $\ell_1$ penalisation is usually controlled by a weight, also called "relaxation parameter", denoted by $\lambda$. It is commonly thought that the practical efficiency of the LASSO for prediction crucially relies on accurate selection of $\lambda$. In this short note, we propose to consider the hyper-parameter selection problem from a new perspective which combines the Hedge online learning method by Freund and Shapire, with the stochastic Frank-Wolfe method for the LASSO. Using the Hedge algorithm, we show that a our simple selection rule can achieve prediction results comparable to Cross Validation at a potentially much lower computational cost.

A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analysis. In this paper, we propose a perturbation bound for the smallest singular value of a given matrix after appending a column, under the assumption that its initial coherence is not large, and we use this bound to derive a fast algorithm for feature extraction., Comment: 14 pages, 6 Figures, Accepted for LVA-ICA 2018 Surrey

This paper studies the complexity of the stochastic gradient algorithm for PCA when the data are observed in a streaming setting. We also propose an online approach for selecting the learning rate. Simulation experiments confirm the practical relevance of the plain stochastic gradient approach and that drastic improvements can be achieved by learning the learning rate., Comment: 11 pages, 1 figure, Submitted to LOD 2018

In a post-prognostics decision context, this paper addresses the problem of maximizing the useful life of a platform composed of several parallel machines under service constraint. Application on multi-stack fuel cell systems is considered. In order to propose a solution to the insufficient durability of fuel cells, the purpose is to define a commitment strategy by determining at each time the contribution of each fuel cell stack to the global output so as to satisfy the demand as long as possible. A relaxed version of the problem is introduced, which makes it potentially solvable for very large instances. Results based on computational experiments illustrate the efficiency of the new approach, based on the Mirror Prox algorithm, when compared with a simple method of successive projections onto the constraint sets associated with the problem.

In this paper, we present a toolbox for a specific optimization problem that frequently arises in bioinformatics or genomics. In this specific optimisation problem, the state space is a set of words of specified length over a finite alphabet. To each word is associated a score. The overall objective is to find the words which have the lowest possible score. This type of general optimization problem is encountered in e.g 3D conformation optimisation for protein structure prediction, or largest core genes subset discovery based on best supported phylogenetic tree for a set of species. In order to solve this problem, we propose a toolbox that can be easily launched using MPI and embeds 3 well-known metaheuristics. The toolbox is fully parametrized and well documented. It has been specifically designed to be easy modified and possibly improved by the user depending on the application, and does not require to be a computer scientist. We show that the toolbox performs very well on two difficult practical problems.

We propose a new procedure for clustering nucleotide sequences based on the "Laplacian Eigenmaps" and Gaussian Mixture modelling. This proposal is then applied to a set of 100 DNA sequences from the mitochondrially encoded NADH dehydrogenase 3 (ND3) gene of a collection of Platyhelminthes and Nematoda species. The resulting clusters are then shown to be consistent with the gene phylogenetic tree computed using a maximum likelihood approach. This comparison shows in particular that the clustering produced by the methodology combining Laplacian Eigenmaps with Gaussian Mixture models is coherent with the phylogeny as well as with the NCBI taxonomy. We also developed a Python package for this procedure which is available online.

The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex optimization over spaces of tensors is now gaining much interest due to its potential applications in signal processing, statistics and engineering. The goal of this paper is to present an extension of the approach by Lewis \cite{lewis1995convex} for the analysis of the subdifferential of certain convex functions of the spectrum of symmetric tensors. We give a complete characterization of the subdifferential of Schatten-type tensor norms for symmetric tensors. Some partial results in this direction are also given for Orthogonally Decomposable tensors.

In the Compressed Sensing community, it is well known that given a matrix $X \in \mathbb R^{n\times p}$ with $\ell_2$ normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence $\mu$ implies a weak RIP, i.e. the singular values of $X_T$ lie between $1-\delta$ and $1+\delta$ for "most" index subsets $T \subset \{1,\ldots,p\}$ with size governed by $\mu$ and $\delta$. In this short note, we show that a small Coherence implies a weak Null Space Property, i.e. $\Vert h_T\Vert_2 \le C \ \Vert h_{T^c}\Vert_1/\sqrt{s}$ for most $T \subset \{1,\ldots,p\}$ with cardinality $|T|\le s$. We moreover prove some singular value perturbation bounds that may also prove useful for other applications.

This paper proposes a variant of the method of Gu\'edon and Verhynin for estimating the cluster matrix in the Mixture of Gaussians framework via Semi-Definite Programming. A clustering oriented embedding is deduced from this estimate. The procedure is suitable for very high dimensional data because it is based on pairwise distances only. Theoretical garantees are provided and an eigenvalue optimisation approach is proposed for computing the embedding. The performance of the method is illustrated via Monte Carlo experiements and comparisons with other embeddings from the literature.

Motivation: LTR retrotransposons are mobile elements that are able, like retroviruses, to copy and move inside eukaryotic genomes. In the present work, we propose a branching model for studying the propagation of LTR retrotransposons in these genomes. This model allows to take into account both positions and degradations of LTR retrotransposons copies. In our model, the duplication rate is also allowed to vary with the degradation level. Results: Various functions have been implemented in order to simulate their spread and visualization tools are proposed. Based on these simulation tools, we show that an accurate estimation of the parameters of this propagation model can be performed. We applied this method to the study of the spread of the transposable elements ROO, GYPSY, and DM412 on a chromosome of \textit{Drosophila melanogaster}. Availability: Our proposal has been implemented using Python software. Source code is freely available on the web at https://github.com/SergeMOULIN/retrotransposons-spread., Comment: 7 pages, 3 figures, 7 tables. Submit to "Bioiformatics" on March 1, 2016

Pinning control on complex dynamical networks has emerged as a very important topic in recent trends of control theory due to the extensive study of collective coupled behaviors and their role in physics, engineering and biology. In practice, real-world networks consists of a large number of vertices and one may only be able to perform a control on a fraction of them only. Controllability of such systems has been addressed in \cite{PorfiriDiBernardo:Automatica08}, where it was reformulated as a global asymptotic stability problem. The goal of this short note is to refine the analysis proposed in \cite{PorfiriDiBernardo:Automatica08} using recent results in singular value perturbation theory., Comment: arXiv admin note: substantial text overlap with arXiv:1406.5441

The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address this problem in the simpler case where $X$ is a random matrix but with the additional constraint that the selected columns be almost orthogonal to a given vector $v$. Our main result is a lower bound on the number of columns we can extract from a normalized i.i.d. Gaussian matrix for the worst $v$., Comment: arXiv admin note: substantial text overlap with arXiv:1203.5223

The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis; see \cite{BourgainTzafriri:IJM87,Tropp:StudiaMath08,Vershynin:IJM01} for methods using random column selection. More constructive approaches have been proposed recently; see the recent contributions of \cite{SpielmanSrivastava:IJM12,Youssef:IMRN14}. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, i.e. a matrix whose singular values all lie in $[1-\epsilon,1+\epsilon]$. We provide individual lower and upper bounds for each singular value of the extracted matrix at the price of conceding only one log factor in the number of columns, when compared to the Restricted Invertibility Theorem of Bourgain and Tzafriri. Our method is fully constructive and the proof is short and elementary., Comment: 5 pages

The problem of estimating ARMA models is computationally interesting due to the nonconcavity of the log-likelihood function. Recent results were based on the convex minimization. Joint model selection using penalization by a convex norm, e.g. the nuclear norm of a certain matrix related to the state space formulation was extensively studied from a computational viewpoint. The goal of the present short note is to present a theoretical study of a nuclear norm penalization based variant of the method of \cite{Bauer:Automatica05,Bauer:EconTh05} under the assumption of a Gaussian noise process.

The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining much interest due to its potential applications to signal processing, statistics and engineering. The goal of this paper is to present an applications to the problem of low rank tensor recovery based on linear random measurement by extending the results of Tropp to the tensors setting., Comment: To appear in proceedings LVA/ICA 2015 at Czech Republic

Sparse recovery from linear Gaussian measurements has been the subject of much investigation since the breaktrough papers \cite{CRT:IEEEIT06} and \cite{donoho2006compressed} on Compressed Sensing. Application to sparse vectors and sparse matrices via least squares penalized with sparsity promoting norms is now well understood using tools such as Gaussian mean width, statistical dimension and the notion of descent cones \cite{tropp2014convex} \cite{Vershynin:ArXivEstimation14}. Extention of these ideas to low rank tensor recovery is starting to enjoy considerable interest due to its many potential applications to Independent Component Analysis, Hidden Markov Models and Gaussian Mixture Models \cite{AnandkumarEtAl:JMLR14}, hyperspectral image analysis \cite{zhang2008tensor}, to name a few. In this paper, we demonstrate that the recent approach of \cite{Vershynin:ArXivEstimation14} provides very useful error bounds in the tensor setting using the nuclear norm or the Romera-Paredes--Pontil \cite{RomeraParedesPontil:NIPS13} penalization.

Given a gene expression data array of a list of bladder cancer patients with their tumor states, it may be difficult to determine which genes can operate as disease markers when the array is large and possibly contains outliers and missing data. An additional difficulty is that observations (tumor states) in the regression problem are discrete ones. In this article, we solve these problems on concrete data using first a clustering approach, followed by Least Absolute Shrinkage and Selection Operator (LASSO) estimators in a nonlinear regression problem involving discrete variables, as described in the brand-new research work of Plan and Vershynin. Gene markers of the most severe tumor state are finally provided using the proposed approach., Comment: submitted to CIBB 2015

The paper studies the expectation of the inspection time in complex aging systems. Under reasonable assumptions, this problem is reduced to studying the expectation of the length of the shortest path in the directed degradation graph of the systems where the parameters are given by a pool of experts. The expectation itself being sometimes out of reach, in closed form or even through Monte Carlo simulations in the case of large systems, we propose an easily computable lower bound. The proposed bound applies to a rather general class of linear programs with random nonnegative costs and is directly inspired from the upper bound of Dyer, Frieze and McDiarmid [Math.Programming {\bf 35} (1986), no.1,3--16].

To extract the relevant features in a given dataset is a difficult task, recently resolved in the non-negative data case with the Non-negative Matrix factorization (NMF) method. The objective of this research work is to extend this method to the case of missing and/or corrupted data due to outliers. To do so, data are denoised, missing values are imputed, and outliers are detected while performing a low-rank non-negative matrix factorization of the recovered matrix. To achieve this goal, a mixture of Bregman proximal methods and of the Augmented Lagrangian scheme are used, in a similar way to the so-called Alternating Direction of Multipliers method. An application to the analysis of gene expression data of patients with bladder cancer is finally proposed.

For two matrices in $\mathbb R^{n_1\times n_2}$, the von Neumann inequality says that their scalar product is less than or equal to the scalar product of their singular spectrum. In this short note, we extend this result to real tensors and provide a complete study of the equality case.

In this paper, we study the perturbation of the extreme singular values of a matrix in the particular case where it is obtained after appending an arbitrary column vector. Such results have many applications in bifurcation theory, signal processing, control theory and many other fields. In the first part of this paper, we review and compare various bounds from recent research papers on this subject. We also present a new lower bound and a new upper bound on the perturbation of the operator norm is provided. Simple proofs are provided, based on the study of the characteristic polynomial rather than on variational methods, as e.g. in \cite{Li-Li}. In a second part of the paper, we present applications to signal processing and control theory.