# Computational approaches to non-convex, sparsity-inducing multi-penalty regularization

## TL;DR

This paper investigates efficient algorithms for non-convex multi-penalty regularization in sparse signal reconstruction, introducing a new infimal convolution approach with proven linear convergence and validated by numerical experiments.

## Contribution

It extends existing methods to non-convex settings, proposes a computationally efficient infimal convolution approach, and provides convergence analysis with numerical validation.

## Key findings

- Both approaches achieve linear convergence rates.
- The infimal convolution method is less dependent on problem size.
- Numerical experiments confirm theoretical convergence rates.

## Abstract

In this work we consider numerical efficiency and convergence rates for solvers of non-convex multi-penalty formulations when reconstructing sparse signals from noisy linear measurements. We extend an existing approach, based on reduction to an augmented single-penalty formulation, to the non-convex setting and discuss its computational intractability in large-scale applications. To circumvent this limitation, we propose an alternative single-penalty reduction based on infimal convolution that shares the benefits of the augmented approach but is computationally less dependent on the problem size. We provide linear convergence rates for both approaches, and their dependence on design parameters. Numerical experiments substantiate our theoretical findings.