On the performance of algorithms for the minimization of $\ell_1$-penalized functionals

TL;DR

This paper evaluates the performance of various algorithms for minimizing -penalized least-squares functionals, introducing a new criterion and comparing five algorithms and two warm-start strategies on different problem conditions.

Contribution

It introduces an approximation isochrone-based criterion and provides a comprehensive comparison of five algorithms and warm-start strategies for -penalized minimization.

Findings

Performance varies significantly between well-conditioned and ill-conditioned problems.

Warm-start strategies improve convergence in certain scenarios.

The introduced criterion effectively assesses algorithm performance.

Abstract

The problem of assessing the performance of algorithms used for the minimization of an $\ell_1$-penalized least-squares functional, for a range of penalty parameters, is investigated. A criterion that uses the idea of `approximation isochrones' is introduced. Five different iterative minimization algorithms are tested and compared, as well as two warm-start strategies. Both well-conditioned and ill-conditioned problems are used in the comparison, and the contrast between these two categories is highlighted.