Stable Recovery of Structured Signals From Corrupted Sub-Gaussian Measurements

TL;DR

This paper presents methods for stable recovery of structured signals from corrupted sub-Gaussian measurements, providing theoretical conditions and regularization guidelines, supported by numerical experiments.

Contribution

It introduces three procedures for signal and corruption reconstruction with theoretical measurement bounds and an extended matrix deviation inequality for sub-Gaussian matrices.

Findings

Conditions for stable recovery derived

Regularization parameters guidance provided

Numerical experiments confirm theoretical results

Abstract

This paper studies the problem of accurately recovering a structured signal from a small number of corrupted sub-Gaussian measurements. We consider three different procedures to reconstruct signal and corruption when different kinds of prior knowledge are available. In each case, we provide conditions (in terms of the number of measurements) for stable signal recovery from structured corruption with added unstructured noise. Our results theoretically demonstrate how to choose the regularization parameters in both partially and fully penalized recovery procedures and shed some light on the relationships among the three procedures. The key ingredient in our analysis is an extended matrix deviation inequality for isotropic sub-Gaussian matrices, which implies a tight lower bound for the restricted singular value of the extended sensing matrix. Numerical experiments are presented to verify our theoretical results.