Robust Matrix Completion

Olga Klopp (), Karim Lounici () and Alexandre Tsybakov ()
Additional contact information
Olga Klopp: CREST, MODAL’X, Université Paris Ouest
Karim Lounici: School of Mathematics, Georgia Institute of Technology
Alexandre Tsybakov: CREST, ENSAE, CNRS

No 2016-14, Working Papers from Center for Research in Economics and Statistics

Abstract: 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 en- tries 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 colum- nwise 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.

Pages: 40
Date: 2016-09
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