Analysis of Max-Product via Local Maxifiers

In this paper we study convergence of the max-product (MP) algorithm on general graphs with cycles. Our analysis follows analogously to that given for the convergence of the sum-product algorithm. We do not work with Gibbs measures but instead we introduce and work with local maxifiers. The contributions of this paper include: reformulation of the MP algorithm on cyclic graphs as max-marginalization on an associated computation tree; existence of local maxifiers and proof that uniqueness of the local maxifier is sufficient for convergence of MP; a Gibbsian theory of local maxifiers and interpretation as operators; an example of non-uniqueness which does not exhibit a phase transition like its Gibbs measure counterpart; and insights into the limitations of Dobrushin-type uniqueness conditions