  Gaussian and quadratic approximations of message passing algorithms on graphs
have attracted considerable recent attention due to their computational
simplicity, analytic tractability, and wide applicability in optimization and
statistical inference problems. This paper presents a systematic framework for
incorporating such approximate message passing (AMP) methods in general
graphical models. The key concept is a partition of dependencies 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
aggregates of many weak edges and integrated with standard message passing
updates on the strong edges. The resulting algorithm, which we call hybrid
generalized approximate message passing (HyGAMP), 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 and multinomial logistic regression
problems are studied as examples of the proposed methodology.
