Theory and Fast Learned Solver for $\ell^1$-TV Regularization

TL;DR

This paper develops a theoretical framework and introduces a fast learned algorithm for $\,\ell^1$-TV regularization, demonstrating improved signal recovery accuracy and efficiency, especially on ECG data.

Contribution

It presents a new mathematical analysis of the $\,\ell^1$-TV model, a novel algorithm PGM-ISTA with proven convergence, and a learned solver LPGM-ISTA that outperforms existing methods.

Findings

LPGM-ISTA achieves higher recovery accuracy.

LPGM-ISTA is more computationally efficient.

Theoretical sample complexity bounds are established.

Abstract

The $\ell^1$ and total variation (TV) penalties have been used successfully in many areas, and the combination of the $\ell^1$ and TV penalties can lead to further improved performance. In this work, we investigate the mathematical theory and numerical algorithms for the $\ell^1$-TV model in the context of signal recovery: we derive the sample complexity of the $\ell^1$-TV model for recovering signals with sparsity and gradient sparsity. Also we propose a novel algorithm (PGM-ISTA) for the regularized $\ell^1$-TV problem, and establish its global convergence and parameter selection criteria. Furthermore, we construct a fast learned solver (LPGM-ISTA) by unrolling PGM-ISTA. The results for the experiment on ECG signals show the superior performance of LPGM-ISTA in terms of recovery accuracy and computational efficiency.