Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions

TL;DR

This paper introduces new optimization algorithms using scale and signed permutation invariant sparsity-promoting functions, demonstrating improved performance in sparse signal recovery and image restoration tasks.

Contribution

It develops and analyzes algorithms incorporating the proximity operator of $( ext{l}_1/ ext{l}_2)^2$ and $ ext{l}_1/ ext{l}_2$, advancing sparse recovery methods with theoretical convergence guarantees.

Findings

Enhanced recovery accuracy in compressed sensing.

Improved computational efficiency under noisy conditions.

Superior performance in high-coherence and high-dynamic-range scenarios.

Abstract

Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of $\ell_1$ to $\ell_2$ norms, $(\ell_1/\ell_2)^2$, has been introduced as a sparsity-prompting function, showing superior performance compared to traditional $\ell_1$ minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of $(\ell_1/\ell_2)^2$ and $\ell_1/\ell_2$ into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions.