Robust Matrix Completion

TL;DR

This paper introduces a convex optimization approach for robust matrix completion that accurately recovers low-rank matrices from incomplete, noisy, and corrupted observations, with theoretical guarantees for optimal recovery rates.

Contribution

It proposes a novel convex estimator combining nuclear norm and sparsity constraints, with rigorous analysis of its recovery guarantees under mixed sampling schemes.

Findings

Guarantees for exact recovery of low-rank and sparse components.

Minimax optimal rates of convergence.

Robustness to noise and corrupted observations.

Abstract

This paper considers the problem of recovery of a low-rank matrix in the situation when most of its entries are not observed and a fraction of observed entries are corrupted. The observations are noisy realizations of the sum of a low rank matrix, which we wish to recover, with a second matrix having a complementary sparse structure such as element-wise or column-wise sparsity. We analyze a class of estimators obtained by solving a constrained convex optimization problem that combines the nuclear norm and a convex relaxation for a sparse constraint. Our results are obtained for the simultaneous presence of random and deterministic patterns in the sampling scheme. We provide guarantees for recovery of low-rank and sparse components from partial and corrupted observations in the presence of noise and show that the obtained rates of convergence are minimax optimal.