A Penalty Function Promoting Sparsity Within and Across Groups

TL;DR

This paper introduces a new weakly-convex penalty function that encourages sparsity both across groups and within individual groups, improving signal processing tasks like denoising and deconvolution.

Contribution

A novel weakly-convex penalty function is proposed, along with associated thresholding methods, for enhanced sparsity promotion in grouped signals.

Findings

Effective in denoising and deconvolution tasks

Outperforms state-of-the-art methods in experiments

Applicable to signals with isolated non-zero features

Abstract

We introduce a new weakly-convex penalty function for signals with a group behavior. The penalty promotes signals with a few number of active groups, where within each group, only a few high magnitude coefficients are active. We derive the threshold function associated with the proposed penalty and study its properties. We discuss how the proposed penalty/threshold function can be useful for signals with isolated non-zeros, such as audio with isolated harmonics along the frequency axis, or reflection functions in exploration seismology where the non-zeros occur on the boundaries of subsoil layers. We demonstrate the use of the proposed penalty/threshold functions in a convex denoising and a non-convex deconvolution formulation. We provide convergent algorithms for both formulations and compare the performance with state-of-the-art methods.