Accurate low-rank matrix recovery from a small number of linear measurements

TL;DR

This paper analyzes the theoretical guarantees of nuclear-norm minimization for recovering low-rank matrices from limited noisy linear measurements, extending previous results to provide near-optimal sample complexity and error bounds.

Contribution

It refines existing theoretical results by establishing near-optimal sample bounds and error guarantees for nuclear-norm minimization in low-rank matrix recovery.

Findings

Nuclear-norm minimization achieves stable recovery with nearly minimal samples.

The method provides order-optimal error bounds in noisy settings.

Theoretical analysis applies to a class of random linear measurements.

Abstract

We consider the problem of recovering a lowrank matrix M from a small number of random linear measurements. A popular and useful example of this problem is matrix completion, in which the measurements reveal the values of a subset of the entries, and we wish to fill in the missing entries (this is the famous Netflix problem). When M is believed to have low rank, one would ideally try to recover M by finding the minimum-rank matrix that is consistent with the data; this is, however, problematic since this is a nonconvex problem that is, generally, intractable. Nuclear-norm minimization has been proposed as a tractable approach, and past papers have delved into the theoretical properties of nuclear-norm minimization algorithms, establishing conditions under which minimizing the nuclear norm yields the minimum rank solution. We review this spring of emerging literature and extend and refine previous theoretical results. Our focus is on providing error bounds when M is well approximated by a low-rank matrix, and when the measurements are corrupted with noise. We show that for a certain class of random linear measurements, nuclear-norm minimization provides stable recovery from a number of samples nearly at the theoretical lower limit, and enjoys order-optimal error bounds (with high probability).