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An efficient proximal algorithm for squared L1 over L2 regularized sparse recovery

Na Zhang, Hong Chen, Qia Li, Junpeng Zhou

TL;DR

The paper tackles sparse recovery under noisy measurements using a squared $L_1/L_2$ regularized model. It introduces a novel proximal algorithm for a general fractional program that includes this model, proves existence of solutions and both subsequential and, under KL conditions, full convergence to critical points. The method yields a closed-form proximal step in the squared $L_1/L_2$ case, enabling efficient iterations and applicability to multiple noise models. Numerical experiments show the approach is competitive in accuracy while delivering substantial speedups over a proximal DC baseline across robust, Cauchy, and Gaussian noise settings.

Abstract

In this paper, we consider a squared $L_1/L_2$ regularized model for sparse signal recovery from noisy measurements. We first establish the existence of optimal solutions to the model under mild conditions. Next, we propose a proximal method for solving a general fractional optimization problem which has the squared $L_1/L_2$ regularized model as a special case. We prove that any accumulation point of the solution sequence generated by the proposed method is a critical point of the fractional optimization problem. Under additional KL assumptions on some potential function, we establish the sequential convergence of the proposed method. When this method is specialized to the squared $L_1/L_2$ regularized model, the proximal operator involved in each iteration admits a simple closed form solution that can be computed with very low computational cost. Furthermore, for each of the three concrete models, the solution sequence generated by this specialized algorithm converges to a critical point. Numerical experiments demonstrate the superiority of the proposed algorithm for sparse recovery based on squared $L_1/L_2$ regularization.

An efficient proximal algorithm for squared L1 over L2 regularized sparse recovery

TL;DR

The paper tackles sparse recovery under noisy measurements using a squared regularized model. It introduces a novel proximal algorithm for a general fractional program that includes this model, proves existence of solutions and both subsequential and, under KL conditions, full convergence to critical points. The method yields a closed-form proximal step in the squared case, enabling efficient iterations and applicability to multiple noise models. Numerical experiments show the approach is competitive in accuracy while delivering substantial speedups over a proximal DC baseline across robust, Cauchy, and Gaussian noise settings.

Abstract

In this paper, we consider a squared regularized model for sparse signal recovery from noisy measurements. We first establish the existence of optimal solutions to the model under mild conditions. Next, we propose a proximal method for solving a general fractional optimization problem which has the squared regularized model as a special case. We prove that any accumulation point of the solution sequence generated by the proposed method is a critical point of the fractional optimization problem. Under additional KL assumptions on some potential function, we establish the sequential convergence of the proposed method. When this method is specialized to the squared regularized model, the proximal operator involved in each iteration admits a simple closed form solution that can be computed with very low computational cost. Furthermore, for each of the three concrete models, the solution sequence generated by this specialized algorithm converges to a critical point. Numerical experiments demonstrate the superiority of the proposed algorithm for sparse recovery based on squared regularization.

Paper Structure

This paper contains 14 sections, 14 theorems, 68 equations, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Solution existence of model \ref{['eq:problem1']}
  4. A proximal algorithm for solving \ref{['eq:problem1']}
  5. First-order optimality condition and the proposed algorithm
  6. The well-definedness of Algorithm \ref{['al:1']}
  7. Subsequential convergence of Algorithm \ref{['al:1']}
  8. Sequential convergence of Algorithm \ref{['al:1']}
  9. KL property
  10. Sequential convergence of Algorithm \ref{['al:1']}
  11. Numerical experiments
  12. Robust compressed sensing problems \ref{['eq:1.4']}.
  13. CS problems with Cauchy noise \ref{['eq:1.3']}.
  14. CS problems with Gaussian noise \ref{['eq:1.2']}.

Key Result

Lemma 3.1

Let $\{x^k : k \in \mathbb{N}\}$ be a minimizing sequence of model eq:problem1. Then any accumulation point of $\{x^k : k \in \mathbb{N}\}$ is an optimal solution of eq:problem1.

Theorems & Definitions (29)

  • Definition 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Theorem 4.1
  • proof
  • Definition 4.1
  • ...and 19 more