Unicity conditions for low-rank matrix recovery

TL;DR

This paper establishes the exact number of measurements needed for low-rank matrix recovery, providing a theoretical benchmark that matches the dimension of the relevant matrix manifold, applicable to any recovery method.

Contribution

It derives necessary and sufficient measurement bounds for uniform and fixed-rank matrix recovery, advancing the theoretical understanding of low-rank matrix recovery.

Findings

m >= 4nr - 4r^2 measurements ensure null space conditions for all rank-r matrices.

m >= 2nr - r^2 + 1 measurements suffice for fixed-rank matrix recovery.

The bounds match the dimension of the rank-2r matrix manifold, serving as a theoretical benchmark.

Abstract

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractible approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m >= Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever --- tractible or not. We show that for a family of random measurement ensembles, m >= 4nr - 4r^2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of $m$ precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m >= 2nr - r^2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization.