Stéphane Chrétien

A Bregman-proximal point algorithm for robust non-negative matrix factorization with possible missing values and outliers - application to gene expression analysis

BackgroundNon-Negative Matrix factorization has become an essential tool for feature extraction in a wide spectrum of applications. In the present work, our objective is to extend the applicability of the method to the case of missing and/or corrupted data due to outliers.ResultsAn essential property for missing data imputation and detection of outliers is that the uncorrupted data matrix is low rank, i.e. has only a small number of degrees of freedom. We devise a new version of the Bregman proximal idea which preserves nonnegativity and mix it with the Augmented Lagrangian approach for simultaneous reconstruction of the features of interest and detection of the outliers using a sparsity promoting ℓ1 penality.ConclusionsAn application to the analysis of gene expression data of patients with bladder cancer is finally proposed.

A clustering package for nucleotide sequences using Laplacian Eigenmaps and Gaussian Mixture Model

In this article, a new Python package for nucleotide sequences clustering is proposed. This package, freely available on-line, implements a Laplacian eigenmap embedding and a Gaussian Mixture Model for DNA clustering. It takes nucleotide sequences as input, and produces the optimal number of clusters along with a relevant visualization. Despite the fact that we did not optimise the computational speed, our method still performs reasonably well in practice. Our focus was mainly on data analytics and accuracy and as a result, our approach outperforms the state of the art, even in the case of divergent sequences. Furthermore, an a priori knowledge on the number of clusters is not required here. For the sake of illustration, this method is applied on a set of 100 DNA sequences taken from the mitochondrially encoded NADH dehydrogenase 3 (ND3) gene, extracted from a collection of Platyhelminthes and Nematoda species. The resulting clusters are tightly consistent with the phylogenetic tree computed using a maximum likelihood approach on gene alignment. They are coherent too with the NCBI taxonomy. Further test results based on synthesized data are then provided, showing that the proposed approach is better able to recover the clusters than the most widely used software, namely Cd-hit-est and BLASTClust.

Enhancing Prony’s method by nuclear norm penalization and extension to missing data

Prony’s method is a widely used method for modelling signals using a finite sum of exponential terms. It has innumerable applications in weather modelling, finance, medical signal analysis, image compression, time series analysis, power grids, etc. Prony’s method has, however, the reputation of being unstable with respect to noise perturbations. The goal of the present paper is to assess the potential improvements of a nuclear-norm-penalized regularization of Prony’s method. The nuclear norm regularization is a standard technique for improving the performance when processing noisy signals with low-rank underlying structure such as in matrix completion, matrix compressed sensing, hidden variable models. We consider both the standard setting and the case of missing data. We provide a fast estimation algorithm for the nuclear-norm-penalized least-squares minimization. Monte Carlo experiments show that regularization can significantly enhance the performance of Prony’s method with and without missing data.

Feature Selection in Weakly Coherent Matrices

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.

A note on computing the smallest conic singular value

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.

Validation of Algorithms to Estimate Distribution Network Characteristics Using Power-Hardware-in-the-Loop Configuration

Distribution system operators (DSOs) require accurate knowledge of the status of the network in order to ensure the continuity and quality of power supply. In this context, the National Physical Laboratory (NPL) and the Power Network Demonstrations Centre (PNDC) have been working together in the development and validation of optimal sensor placement and network topology estimation algorithms. This paper presents the description of two of these algorithms as well as the topology configuration of the PNDC distribution network considered to gather measurements for the validation of the algorithms. A Power-Hardware-in-the-Loop (P-HiL) configuration has been used as the testbed, where a number of physical measurement devices are installed in the physical network and an extended number of devices are virtually installed in the simulate network. The applications of the proposed algorithms to the measurements along with results from the P-HiL tests are presented in the paper.

An Elementary Approach to the Problem of Column Selection in a Rectangular Matrix

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

Simulation-based estimation of branching models for LTR retrotransposons

[1-\epsilon,1+\epsilon]

Corrigendum to “A note on computing the Smallest Conic Singular Value” [J. Comput. Appl. Math. 340 (2018) 221–230]

. 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.

Application of Robust PCA with a structured outlier matrix to topology estimation in power grids

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 us to take into account both the positions and the degradation level 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 have developed a first method to evaluate the parameters of this propagation model. We applied this method to the study of the spread of the transposable elements ROO, GYPSY and DM412 on a chromosome of Drosophila melanogaster. Availability and Implementation: Our proposal has been implemented using Python software. Source code is freely available on the web at https://github.com/SergeMOULIN/retrotransposons‐spread. Contact: serge.moulin@univ‐fcomte.fr Supplementary information: are available at Bioinformatics online.