2005/08/15 by Martin J. Wainwright, Tommi Jaakkola, Wainwright, Martin J. +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · #Algorithms and Data Compression #Artificial Intelligence (cs.AI) #DNA and Biological Computing #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.cs/0508070
openalex publication_date 2005/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop and analyze methods for computing provably optimal \em maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a \em tree-relaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a \em tree-reweighted max-product message-passing algorithm that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the mode-finding problem, and a reweighted form of the max-product (min-sum) message-passing algorithm.