Matrix completion by singular value thresholding: sharp bounds

TL;DR

This paper provides strong theoretical guarantees for an iterative thresholding method in matrix completion, achieving minimax optimal rates of convergence and improving understanding of its performance.

Contribution

It offers the first sharp theoretical bounds for an iterative thresholding algorithm, matching those of nuclear-norm penalization methods.

Findings

Achieves exact minimax optimal rates of convergence.

Provides theoretical guarantees for a modified softImpute algorithm.

Bridges the gap between empirical success and theoretical understanding.

Abstract

We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is a modification of the softImpute algorithm. An im-portant consequence of our result is the exact minimax optimal rates of convergence for matrix completion problem which were known until know only up to a logarithmic factor.