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

Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of $ℓ_1$ to $ℓ_2$ norms, $(ℓ_1/ℓ_2)^2,$ has been introduced as a sparsity-prompting function, showing superior performance compared to traditional $ℓ_1$ minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of $(ℓ_1/ℓ_2)^2$ and $ℓ_1/ℓ_2$ 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.