1Hybrid Approximate Message Passing with Applications to Structured Sparsity

Abstract—Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractabil-ity, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incor-porating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of depen-dencies of a general graphical model into strong and weak edges, with the weak edges representing interactions through aggregates of small, linearizable couplings of variables. AMP approximations based on the Central Limit Theorem can be readily applied to the weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (Hybrid-GAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance–complexity trade-off can be achieved. Group sparsity problems are studied as an example of this general methodology where there is a natural partition of edges. Index Terms—belief propagation, estimation, group sparsity, max-sum algorithm, maximum a posteriori probability, minimum mean-squared error, optimization, simultaneous sparsity, sum-product algorithm I