Fixed Points of Loopy Belief Propagation as Zero Gradients of a Function of Reparameterizations

Abstract

The existing view on loopy belief propagation sees it as an algorithm to nd a common zero of a system of non-linear functions, not explicitly related to each other. We show that these functions are in fact related { they are the partial derivatives of a single function of reparameterizations. Thus, belief propagation searches for a zero gradient of a single function. We show that belief propagation xed points are in one-to-one correspondence with zero gradient points of this function and that every such point is a saddle.