Marginal Consistency: Upper-Bounding Partition Functions over Commutative Semirings

IEEE Transactions on Pattern Analysis and Machine Intelligence
Tomás Werner

Abstract

Many inference tasks in pattern recognition and artificial intelligence lead to partition functions in which addition and multiplication are abstract binary operations forming a commutative semiring. By generalizing max-sum diffusion (one of convergent message passing algorithms for approximate MAP inference in graphical models), we propose an iterative algorithm to upper bound such partition functions over commutative semirings. The iteration of the algorithm is remarkably simple: change any two factors of the partition function such that their product remains the same and their overlapping marginals become equal. In many commutative semirings, repeating this iteration for different pairs of factors converges to a fixed point when the overlapping marginals of every pair of factors coincide. We call this state marginal consistency. During that, an upper bound on the partition function monotonically decreases. This abstract algorithm unifies several existing algorithms, including max-sum diffusion and basic constraint propagation (or local consistency) algorithms in constraint programming. We further construct a hierarchy of marginal consistencies of increasingly higher levels and show than any such level can be enforced by addi...Continue Reading

References

Sep 22, 2006·IEEE Transactions on Pattern Analysis and Machine Intelligence·Vladimir Kolmogorov
May 15, 2007·IEEE Transactions on Pattern Analysis and Machine Intelligence·Tomás Werner
Apr 19, 2008·IEEE Transactions on Pattern Analysis and Machine Intelligence·Richard SzeliskiCarsten Rother
May 19, 2010·IEEE Transactions on Pattern Analysis and Machine Intelligence·Nikos KomodakisGeorgios Tziritas
Jun 19, 2010·IEEE Transactions on Pattern Analysis and Machine Intelligence·Tomás Werner
Feb 1, 1979·IEEE Transactions on Pattern Analysis and Machine Intelligence·R M Haralick, L G Shapiro

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