l1-Norm Minimization with Regula Falsi Type Root Finding Methods

TL;DR

This paper introduces an efficient, derivative-free Regula Falsi root-finding approach to solve nonconvex level-set formulations for sparse inverse problems, extending traditional convex methods to broader nonconvex likelihoods.

Contribution

It develops a novel application of Regula Falsi methods for nonconvex likelihoods in level-set formulations, enabling robust sparse solutions beyond convex constraints.

Findings

Effective extension of level-set methods to nonconvex problems

Demonstrated practical performance with l1-regularized Student's t inversion

Provides a simple, derivative-free root-finding technique for complex inverse problems

Abstract

Sparse level-set formulations allow practitioners to find the minimum 1-norm solution subject to likelihood constraints. Prior art requires this constraint to be convex. In this letter, we develop an efficient approach for nonconvex likelihoods, using Regula Falsi root-finding techniques to solve the level-set formulation. Regula Falsi methods are simple, derivative-free, and efficient, and the approach provably extends level-set methods to the broader class of nonconvex inverse problems. Practical performance is illustrated using l1-regularized Student's t inversion, which is a nonconvex approach used to develop outlier-robust formulations.