Nearly-optimal Robust Matrix Completion

TL;DR

This paper introduces a simple projected gradient descent algorithm for robust matrix completion that achieves nearly optimal sample complexity and corruption tolerance, significantly improving computational efficiency over existing methods.

Contribution

The paper presents a novel, nearly linear time algorithm for robust matrix completion and robust PCA, with theoretical guarantees and empirical validation.

Findings

Achieves nearly optimal sample complexity and corruption tolerance.

Significantly improves time complexity over previous methods.

Empirically outperforms existing algorithms by an order of magnitude.

Abstract

In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple projected gradient descent method to estimate the low-rank matrix that alternately performs a projected gradient descent step and cleans up a few of the corrupted entries using hard-thresholding. Our algorithm solves RMC using nearly optimal number of observations as well as nearly optimal number of corruptions. Our result also implies significant improvement over the existing time complexity bounds for the low-rank matrix completion problem. Finally, an application of our result to the robust PCA problem (low-rank+sparse matrix separation) leads to nearly linear time (in matrix dimensions) algorithm for the same; existing state-of-the-art methods require quadratic time. Our empirical results corroborate our theoretical results and show that even for moderate sized problems, our method for robust PCA is an an order of magnitude faster than the existing methods.