Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization

TL;DR

This paper analyzes noisy low-rank matrix completion, demonstrating that convex relaxation achieves near-optimal estimation errors under certain conditions by connecting it with the nonconvex Burer-Monteiro approach, explaining its empirical success.

Contribution

It bridges convex relaxation and nonconvex optimization to provide theoretical guarantees for noisy matrix completion, explaining practical effectiveness.

Findings

Convex relaxation achieves near-optimal errors for bounded rank and condition number.

Approximate critical points of nonconvex formulation closely approximate convex solutions.

The approach is robust against various noise levels.

Abstract

This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-\`a-vis random noise. When the rank and the condition number of the unknown matrix are bounded by a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors --- in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss --- for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, thus allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.