Optimal Sample Complexity for Stable Matrix Recovery

TL;DR

This paper establishes the fundamental limits on the number of measurements needed for stable recovery of structured matrices, such as sparse or low-rank matrices, across various measurement models, filling a key theoretical gap.

Contribution

It provides the first optimal sample complexity bounds for stable matrix recovery applicable to multiple measurement types and structures, unifying the theory under a common framework.

Findings

Optimal sample complexity bounds are proven for stable matrix recovery.

Recovery is stable with high probability under various measurement models.

The results apply to matrices with sparsity, low-rankness, and other parsimonious structures.

Abstract

Tremendous efforts have been made to study the theoretical and algorithmic aspects of sparse recovery and low-rank matrix recovery. This paper fills a theoretical gap in matrix recovery: the optimal sample complexity for stable recovery without constants or log factors. We treat sparsity, low-rankness, and potentially other parsimonious structures within the same framework: constraint sets that have small covering numbers or Minkowski dimensions. We consider three types of random measurement matrices (unstructured, rank-1, and symmetric rank-1 matrices), following probability distributions that satisfy some mild conditions. In all these cases, we prove a fundamental result -- the recovery of matrices with parsimonious structures, using an optimal (or near optimal) number of measurements, is stable with high probability.