Restricted strong convexity and weighted matrix completion: Optimal bounds with noise

TL;DR

This paper establishes optimal error bounds for weighted matrix completion under noisy sampling, leveraging restricted strong convexity and less restrictive matrix conditions, and proves these bounds are nearly optimal.

Contribution

It introduces a new analysis of the observation operator satisfying restricted strong convexity for weighted sampling, leading to optimal error bounds with relaxed matrix conditions.

Findings

Error bounds are established in weighted Frobenius norm under noise.

The method handles matrices with less restrictive 'spikiness' and 'low-rankness' conditions.

The bounds are shown to be nearly optimal via information-theoretic lower bounds.

Abstract

We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the "spikiness" and "low-rankness" of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an $M$-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with $\ell_q$-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.