Entropy Minimization for Convex Relaxation Approaches

Despite their enormous success in solving hard combinatorial problems, convex relaxation approaches often suffer from the fact that the computed solutions are far from binary and that subsequent heuristic binarization may substantially degrade the quality of computed solutions. In this paper, we propose a novel relaxation technique which incorporates the entropy of the objective variable as a measure of relaxation tightness. We show both theoretically and experimentally that augmenting the objective function with an entropy term gives rise to more binary solutions and consequently solutions with a substantially tighter optimality gap. We use difference of convex function (DC) programming as an efficient and provably convergent solver for the arising convex-concave minimization problem. We evaluate this approach on three prominent non-convex computer vision challenges: multi-label inpainting, image segmentation and spatio-temporal multi-view reconstruction. These experiments show that our approach consistently yields better solutions with respect to the original integral optimization problem.