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High-dimensional robust structure learning


Directeur de thèse :     Sophie ACHARD

École doctorale : Electronique, electrotechnique, automatique, traitement du signal (EEATS)

Spécialité : Signal, image, parole, télécoms

Structure de rattachement : Université Grenoble Alpes

Établissement d'origine : INPG/ENSIMAG

Financement(s) : Contrat doctoral ; Contrat doctoral


Date d'entrée en thèse : 01/10/2016

Date de soutenance : 20/12/2019


Composition du jury :
> Stéphanie ALLASSONNIERE, Professeur, Paris Descartes University, Rapporteur
Monsieur Frédéric RICHARD Professeur, Université Aix-Marseille , Rapporteur
Monsieur Paulo GONCALVES, Directeur de Recherche, ENS Lyon, Examinateur
Monsieur Ronald PHLYPO Maître de Conférences, Grenoble INP, Examinateur
Madame Sophie ACHARD, Directeur de Recherche, CNRS, Directeur de thèse
Madame Florence FORBES, Directeur de Recherche, INRIA, Co-directeur de thèse


Résumé : Structure learning in graphical models is an essential topic in different application areas, i.e., genetics, neuroscience. The crucial part of this model is the estimation of covariance/precision matrices. Traditional techniques for handling this problem suffer from two main issues. The first one is the lack of robustness when samples are assumed to follow a Gaussian distribution. The second one is the lack of data when the number of parameters to estimate is too large compared to the number of samples. Thus this thesis aims to build robust high-dimensional models for covariance and precision matrices estimation. In the first main contribution of this thesis we consider the shrinked likelihood-based estimators of the covariance matrix under the assumption of heavy-tailed distribution with unknown mean vector. The main difficulty at this point is the choice of the regularization parameters. We provide a closed-form expression of an optimal shrinkage coefficient for any sample distribution in the elliptical family. Based on these results, an algorithm for the case of the multivariate t-distribution with the simulated and real data is presented. The second contribution is dealing with sparse precision matrix estimation for the non-Gaussian data. Starting with the traditional techniques, we are able to generalize results for the high-dimensional mixture models for the subclass of elliptical family. Finally, we test our graph structure learning approach on brain signals using fMRI. The structure induced by both the correlation and the partial correlation is considered. We then propose a new graph construction method taking into account both conditional and marginal independences. The proposed approach shows better results than classical algorithms.

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