This paper considers the problem of estimation of a low-rank matrix when most of its entries are not observed and some of the observed entries are corrupted. The observations are noisy realizations of a sum of a low-rank matrix, which we wish to estimate, and a second matrix having a complementary sparse structure such as elementwise sparsity or columnwise sparsity. We analyze a class of estimators obtained as solutions of a constrained convex optimization problem combining the nuclear norm penalty and a convex relaxation penalty for the sparse constraint. Our assumptions allow for simultaneous presence of random and deterministic patterns in the sampling scheme. We establish rates of convergence for the low-rank component from partial and corrupted observations in the presence of noise and we show that these rates are minimax optimal up to logarithmic factors. Lien vers l'article
KLOPP, O., LOUNICI, K. and TSYBAKOV, A. (2017). Robust Matrix Completion. Probability Theory and Related Fields, 169(43862), pp. 523-564.