Stable Recovery of Regularized Linear Inverse Problems

TL;DR

This work provides a complete second‑order geometric characterization of stable recovery for convex regularized linear inverse problems, showing stability is equivalent to a nontrivial intersection condition involving the kernel of the forward operator and a tangent cone to the subdifferential of the Fenchel conjugate of the regularizer. It extends stable recovery theory beyond polyhedral settings, including analysis group sparsity and isotropic total variation, and yields verifiable conditions that guarantee linear convergence rates even when sharp minima fail. The authors also demonstrate their theory through numerical experiments on group sparsity and TV, illustrating practical applicability and guiding future extensions to nuclear norm and RIP‑based analyses. Overall, the paper advances a deeper understanding of when regularized inverse problems yield reliably stable solutions under small perturbations.

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.

Figures (4)

• Figure 1: Proportion of cases for which $x_0$ can be recovered as a function of $m$.
• Figure 2: Group sparsity problems with different active groups of the signals
• Figure 3: Images randomly sampled from the Extended MNIST dataset
• Figure 4: Isotropic total variation problems with different group sparsity of the images' gradients.

Theorems & Definitions (35)

• Definition 2.1: Tangent/contingent cone
• Definition 2.2: Sharp and strong minima
• Definition 2.3: directional derivative and second subderivative
• Lemma 2.4: Characterizations for sharp and strong solutions
• Definition 3.1: Stable recovery
• Example 3.2: Stable recovery without sharp minima
• Theorem 3.3: Geometric characterization of stable recovery
• proof
• Example 3.4: Failure of stable recovery under strong minima
• Proposition 3.5