l1-Norm Minimization with Regula Falsi Type Root Finding Methods

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.