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Modèles bayésiens pour l’identification de représentations antiparcimonieuses et l’analyse en composantes principales bayésienne non paramétrique

Abstract : This thesis proposes Bayesian parametric and nonparametric models for signal representation. The first model infers a higher dimensional representation of a signal for sake of robustness by enforcing the information to be spread uniformly. These so called anti-sparse representations are obtained by solving a linear inverse problem with an infinite-norm penalty. We propose in this thesis a Bayesian formulation of anti-sparse coding involving a new probability distribution, referred to as the democratic prior. A Gibbs and two proximal samplers are proposed to approximate Bayesian estimators. The algorithm is called BAC-1. Simulations on synthetic data illustrate the performances of the two proposed samplers and the results are compared with state-of-the art methods. The second model identifies a lower dimensional representation of a signal for modelisation and model selection. Principal component analysis is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Few works have proposed a probabilistic approach to infer the number of significant components. We propose a Bayesian nonparametric principal component analysis called BNP-PCA. The proposed model involves an Indian buffet process to promote a parsimonious use of principal components, which is assigned a prior distribution defined on the manifold of orthonormal basis. Inference is done using MCMC methods. The estimators of the latent dimension are theoretically and empirically studied. The relevance of the approach is assessed on two applications
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https://tel.archives-ouvertes.fr/tel-01730077
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Submitted on : Thursday, May 3, 2018 - 4:30:07 PM
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Clément Elvira. Modèles bayésiens pour l’identification de représentations antiparcimonieuses et l’analyse en composantes principales bayésienne non paramétrique. Traitement du signal et de l'image [eess.SP]. Ecole Centrale de Lille, 2017. Français. ⟨NNT : 2017ECLI0016⟩. ⟨tel-01730077v2⟩

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