Stable Recovery of Regularized Linear Inverse Problems

TL;DR

This paper introduces new theoretical characterizations for stable recovery in regularized linear inverse problems, emphasizing the role of nonsmooth second-order information and providing practical conditions for group sparsity and total variation problems.

Contribution

It offers novel finite-dimensional stable recovery characterizations and practical conditions for analysis group sparsity and total variation problems.

Findings

New sufficient conditions for stable recovery of group sparsity.

Numerical experiments confirm the effectiveness of the proposed conditions.

Deeper understanding of nonsmooth second-order information in inverse problems.

Abstract

Recovering a low-complexity signal from its noisy observations by regularization methods is a cornerstone of inverse problems and compressed sensing. Stable recovery ensures that the original signal can be approximated linearly by optimal solutions of the corresponding Morozov or Tikhonov regularized optimization problems. In this paper, we propose new characterizations for stable recovery in finite-dimensional spaces, uncovering the role of nonsmooth second-order information. These insights enable a deeper understanding of stable recovery and their practical implications. As a consequence, we apply our theory to derive new sufficient conditions for stable recovery of the analysis group sparsity problems, including the group sparsity and isotropic total variation problems. Numerical experiments on these two problems give favorable results about using our conditions to test stable recovery.