Optimal sample complexity for stable matrix recovery

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.

Blind Gain and Phase Calibration via Sparse Spectral Methods

“…To position BGPC in a more broad context, it is a special bilinear inverse problem [5], which in turn is a special case of low-rank matrix recovery from incomplete measurements [24], [25], [26], [27]. A resurgence of interest in bilinear inverse problems was pioneered by the recent studies in single-channel blind deconvolution of signals with subspace or sparsity structures, where both the signal and the filter are structured [28], [29], [30], [31], [32].…”