Optimal Sample Complexity for Stable Matrix Recovery

TL;DR

This paper establishes the minimal number of measurements needed for stable recovery of low-rank and sparse matrices across various measurement types, filling a key theoretical gap in matrix recovery.

Contribution

It provides a unified theoretical framework for the optimal sample complexity in stable matrix recovery without extra constants or log factors.

Findings

Proves stability of matrix recovery with minimal measurements across different measurement models.

Establishes near-optimal sample complexity bounds for structured matrix recovery.

Unifies analysis for sparse, low-rank, 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.