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

Xinling Liu, Jianjun Wang, Bangti Jin

TL;DR

This work advances the theory and computation for recovering signals from compressed measurements using the combined $\ell^1$ and TV penalties ($\ell^1$-TV). It derives a robust recovery guarantee by bounding the descent-cone-related statistical dimension with a closed-form bound $\Phi(s_r,s_g)$ that depends on the sparsity $s_r$ and gradient sparsity $s_g$, and translates this into a sampling requirement $m>(\sqrt{\Phi(s_r,s_g)}+t)^2+1$ under Gaussian measurements. It then develops a proximal-gradient-mapping-based solver (PGM-ISTA) with global convergence guarantees and a practical parameter-selection strategy, and augments it with LPGM-ISTA by unrolling the algorithm into a learned network for faster performance. Numerical experiments, including synthetic sensing and ECG signal recovery, demonstrate that LPGM-ISTA achieves superior accuracy with substantially reduced computation compared to conventional methods. Overall, the paper provides a rigorous recovery theory for a fused regularizer and a scalable, data-driven solver that is particularly effective for signals with simultaneous sparsity and gradient sparsity.

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.

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

TL;DR

This work advances the theory and computation for recovering signals from compressed measurements using the combined and TV penalties (-TV). It derives a robust recovery guarantee by bounding the descent-cone-related statistical dimension with a closed-form bound that depends on the sparsity and gradient sparsity , and translates this into a sampling requirement under Gaussian measurements. It then develops a proximal-gradient-mapping-based solver (PGM-ISTA) with global convergence guarantees and a practical parameter-selection strategy, and augments it with LPGM-ISTA by unrolling the algorithm into a learned network for faster performance. Numerical experiments, including synthetic sensing and ECG signal recovery, demonstrate that LPGM-ISTA achieves superior accuracy with substantially reduced computation compared to conventional methods. Overall, the paper provides a rigorous recovery theory for a fused regularizer and a scalable, data-driven solver that is particularly effective for signals with simultaneous sparsity and gradient sparsity.

Abstract

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

Paper Structure

This paper contains 18 sections, 12 theorems, 95 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recovery guarantee for $\ell^1$-TV regularization
  4. Fast learned solver for regularized $\ell^1$-TV model
  5. Proximal gradient mapping method
  6. Construction of proximal gradient mapping method
  7. Global convergence and parameter selection criterion for PGM-ISTA
  8. Construction of fast learned solver
  9. Numerical experiments and discussions
  10. Synthetic experiments for sampling number
  11. Parameters influence for PGM-ISTA
  12. LPGM-ISTA for recovering ECG signals
  13. Technical proofs
  14. Proof of Theorem \ref{['l1tv-theorem3.1']}
  15. Proof of Theorem \ref{['l1tv-theorem3.2']}
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\mathbb{K}$ be a convex cone. Then the statistical dimension $\delta(\mathbb{K})$ and the conic Gaussian width $w(\mathbb{K})$ satisfy

Figures (9)

  • Figure 1: The comparison of $\ell^1$, TV and $\ell^1$-TV under Gaussian sampling with a sampling ratio of 0.5.
  • Figure 2: For fixed $\lambda_2 = 1$, plots of $\Phi(s_r,s_g)$ ($s_g\leq2s_r$) under multiple settings of $n$ and $\lambda_1$.
  • Figure 3: Neural network representation of PGM-ISTA. (a) PGM-ISTA; (b) LPGM-ISTA obtained by unfolding PGM-ISTA with $L$ layers, and the learnable parameters are $\Theta^L=\{\boldsymbol{W}_{\boldsymbol{x}}, \boldsymbol{W}_{\boldsymbol{y}},u,t\}$.
  • Figure 4: A synthetic signal and its recovered one by the $\ell^1$-TV method.
  • Figure 5: Selections of $\lambda_1$ for fixed $\lambda_2=1$ under different situations.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Definition 2.1: tropp2015convex
  • Definition 2.2: rockafellar2015convex
  • Definition 2.3: rockafellar2015convex
  • Definition 2.4: amelunxen2014living
  • Definition 2.5: tropp2015convex
  • Lemma 2.1: amelunxen2014living
  • Definition 2.6: beck2017first
  • Definition 2.7: beck2017first
  • Lemma 2.2: liu2010an
  • Theorem 3.1
  • ...and 21 more