An Evolutionary Multiobjective Approach to Sparse Reconstruction

Abstract

This paper addresses the problem of finding sparse solutions to linear systems. Although this problem involves two competing cost function terms (measurement error and a sparsity-inducing term), previous approaches combine these into a single cost term and solve the problem using conventional numerical optimization methods. In contrast, the main contribution of this paper is to use a multiobjective approach. The paper begins by investigating the sparse reconstruction problem, and presents data to show that knee regions do exist on the Pareto front (PF) for this problem and that optimal solutions can be found in these knee regions. Another contribution of the paper, a new soft-thresholding evolutionary multiobjective algorithm (StEMO), is then presented, which uses a soft-thresholding technique to incorporate two additional heuristics: one with greater chance to increase speed of convergence toward the PF, and another with higher probability to improve the spread of solutions along the PF, enabling an optimal solution to be found in the knee region. Experiments are presented, which show that StEMO significantly outperforms five other well known techniques that are commonly used for sparse reconstruction. Practical applications are also demonstrated to fundamental problems of recovering signals and images from noisy data.