Theory and fast learned solver for ℓ 1 -TV regularization

Abstract The ℓ 1 and total variation (TV) penalties have been used successfully in many areas, and the combination of the ℓ 1 and TV penalties can lead to further improved performance. In this work, we investigate the mathematical theory and numerical algorithms for the ℓ 1 -TV model in the context of signal recovery: we derive the sample complexity of the ℓ 1 -TV model for recovering signals with sparsity and gradient sparsity. Also we propose a novel algorithm (PGM-ISTA) for the regularized ℓ 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.