Majorize–Minimize Adapted Metropolis–Hastings Algorithm

The dimension and the complexity of inference problems have dramatically increased in statistical signal processing. It thus becomes mandatory to design improved proposal schemes in Metropolis-Hastings algorithms, providing large proposal transitions that are accepted with high probability. The proposal density should ideally provide an accurate approximation to the target density with a low computational cost. In this paper, we derive a novel Metropolis-Hastings proposal, inspired from Langevin dynamics, where the drift term is preconditioned by an adaptive matrix constructed through a Majorization-Minimization strategy. We propose several variants of low-complexity curvature metrics applicable to large scale problems. We demonstrate the geometric ergodicity of the resulting chain for the class of super-exponential distributions. The proposed method is shown to exhibit a good performance in two signal recovery examples.