l1-Norm Minimization With Regula Falsi Type Root Finding Methods

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. Extending these approaches to nonconvex likelihood constraints enables outlier robust methods. 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. The approach provably extends level-set methods to the broader class of nonconvex inverse problems. Practical performance is illustrated using l(1)-regularized Student's t inversion, which is a nonconvex problem used to develop outlier-robust approaches.