Complétion de matrice : aspects statistiques et computationnels

Abstract : This thesis deals with the low rank matrix completion methods and focuses on some related problems, of both statistical and algorithmic nature. The first part of this work extends the existing statistical guarantees obained for sub-Gaussian additive noise models, to more general distributions. In particular,we provide upper bounds on the prediction error of trace norm penalized estimatorwith high probability for multinomial distributions and for distributions belonging to the exponential family. For the latter, we prove that the trace norm penalized estimators are minimax optimal up to a logarithmic factor by giving a lower bound.The second part of this work focuses on the conditionnal gradient algorithm, which is used in particular to compute previous estimators. We consider the stochastic optimization framework and gives the convergence rate of twovariants of the conditional gradient algorithm. We gives the conditions under which these algorithms match the performance of projected gradient algorithms.
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Jean Lafond. Complétion de matrice : aspects statistiques et computationnels. Statistiques [math.ST]. Université Paris-Saclay, 2016. Français. ⟨NNT : 2016SACLT002⟩. ⟨tel-01529861⟩

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