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一般化された確率伝搬法の数学的構造

一般化された確率伝搬法の数学的構造. 東北大学大学院情報科学研究科 田中和之 kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/. Factor Graph and Loopy Belief Propagation F. R. Kschischang, B. J. Frey and H.-A. Loeliger: IEEE Transactions on information theory, vol. 47, no. 2, 2001. Introduction.

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一般化された確率伝搬法の数学的構造

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  1. 一般化された確率伝搬法の数学的構造 東北大学大学院情報科学研究科 田中和之 kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ DEX-SMI2008

  2. Factor Graph and Loopy Belief Propagation • F. R. Kschischang, B. J. Frey and H.-A. Loeliger: IEEE Transactions on information theory, vol. 47, no. 2, 2001. Introduction • Generalized Belief Propagation • J. S. Yedidia, W. T. Freeman and Y. Weiss: IEEE Transactions on Information Theory, vol.51, no.7, pp.2282-2312, 2005. • Simplified Cluster Variation Method • T. Morita: Physica A, vol.155, no.1, pp.73-83, 1989. Factor Graph による確率伝搬法の表現を一般化された確率伝搬法(Generalized Belief Propagation)とSimplified Cluster Variation Methodの定式化の立場から導出する． DEX-SMI2008

  3. 1 2 1 2 1 2 3 4 3 4 3 4 Generalized Belief Propagation Cluster: Set of nodes Every subcluster of the element ofCdoes not belong toC. Example: System consisting of 4 nodes DEX-SMI2008

  4. 1 2 1 2 1 2 3 4 3 4 3 4 Generalized Belief Propagation KL Divergence Sum of all the Basic Clusters and their Sub-Clusters Example Subclusters Basic Clusters DEX-SMI2008

  5. 1 2 3 4 5 6 7 8 9 5 2 1 4 5 3 6 2 2 1 2 3 4 1 3 6 5 2 6 5 8 9 8 7 4 5 5 4 7 8 9 6 5 8 7 4 8 6 5 9 Selection of Basic Cluster in LBP and GBP LBP (Bethe Approx.） GBP (Square Approx. in CVM; Kikuchi Approx.) DEX-SMI2008

  6. V: すべての頂点の集合 Generalized Belief Propagation : 3個の頂点からなるハイパエッジの集合 : C に属するハイパエッジの副クラスターの集合 B の要素はノードとエッジのみ : Bに属するのエッジの副クラスターの集合 A の要素はノードのみ DEX-SMI2008

  7. 5 2 5 2 2 5 1 1 1 1 1 4 3 5 4 3 2 4 3 1 1 1 1 4 3 Generalized Belief Propagation 1 DEX-SMI2008

  8. 5 5 5 2 2 2 5 5 5 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 4 4 4 3 3 3 Generalized Belief Propagation 1 1 1 DEX-SMI2008

  9. + + + 5 5 5 2 2 2 - - - - + 5 2 1 2 5 5 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 4 3 1 4 3 4 3 1 Reducibility Conditions Generalized Belief Propagation Marginal Probabilities are determined so as to minimize the Kikuchi Free Energy under some Reducibility Conditions. ~ = Kikuchi Free Energy DEX-SMI2008

  10. 5 2 2 1 1 4 3 1 1 5 2 3 1 5 4 3 1 4 3 1 4 Generalized Belief Propagation DEX-SMI2008

  11. 2 5 1 1 5 5 2 2 1 1 4 3 2 5 1 1 4 3 1 1 1 4 3 4 3 Generalized Belief Propagation DEX-SMI2008

  12. 5 5 5 5 2 2 2 2 4 3 4 3 1 1 1 1 4 3 4 3 Generalized Belief Propagation DEX-SMI2008

  13. 5 5 5 5 2 2 2 2 4 3 1 1 1 1 4 3 4 3 Generalized Belief Propagation 4 3 DEX-SMI2008

  14. 5 5 5 5 2 2 2 2 4 3 1 1 1 1 4 3 Generalized Belief Propagation 4 3 4 3 DEX-SMI2008

  15. 4 3 5 5 5 5 5 5 2 2 2 2 2 2 1 1 1 1 1 1 4 3 4 3 4 3 4 3 4 3 Generalized Belief Propagation DEX-SMI2008

  16. + + + 5 5 5 2 2 2 - - - - + 5 2 1 2 5 5 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 4 3 4 3 4 3 Generalized Belief Propagation Marginal Probabilities are determined so as to minimize the Kikuchi Free Energy under some Reducibility Conditions. ~ = Kikuchi Free Energy 1 1 Reducibility Conditions DEX-SMI2008

  17. + + + 5 5 5 2 2 2 - - - - + 5 2 1 2 5 5 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 4 3 4 3 4 3 Simplified Cluster Variation Method Marginal Probabilities are determined so as to minimize the Kikuchi Free Energy under some Reducibility Conditions. ~ = 1 1 Reducibility Conditions DEX-SMI2008

  18. 5 2 2 1 1 4 3 1 1 5 2 3 1 5 4 3 1 4 3 1 4 Simplified Cluster Variation Method DEX-SMI2008

  19. 2 5 1 1 5 5 2 2 1 1 4 3 2 5 1 1 4 3 1 1 1 4 3 4 3 Simplified Cluster Variation Method DEX-SMI2008

  20. 5 5 5 5 2 2 2 2 4 3 4 3 1 1 1 1 4 3 Simplified Cluster Variation Method 4 3 DEX-SMI2008

  21. 4 3 5 5 5 5 5 5 2 2 2 2 2 2 1 1 1 1 1 1 4 3 4 3 4 3 4 3 4 3 Simplified Cluster Variation Method DEX-SMI2008

  22. 5 2 1 5 2 5 2 2 5 1 4 3 1 1 1 1 4 3 5 4 3 2 4 3 1 1 1 1 4 3 Simplified Cluster Variation Methodand Factor Graph 1 DEX-SMI2008

  23. 5 2 5 2 2 5 1 1 1 1 1 4 3 5 4 3 2 4 3 1 1 1 1 4 3 Generalized Belief Propagation 1 DEX-SMI2008

  24. 5 2 5 2 2 5 1 1 1 1 1 4 3 5 4 3 2 4 3 1 1 1 1 4 3 Generalized Belief Propagation 1 DEX-SMI2008

  25. 5 5 5 2 2 2 5 2 2 5 5 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 4 3 1 4 3 4 3 1 Reducibility Conditions Generalized Belief Propagation Marginal Probabilities are determined so as to minimize the Kikuchi Free Energy under some Reducibility Conditions. ~ + + + = - - - - + 1 Kikuchi Free Energy DEX-SMI2008

  26. 5 5 5 2 2 2 5 2 2 5 1 1 1 1 1 1 1 4 3 4 3 2 1 4 3 Generalized Belief Propagation Marginal probabilities are determined so as to minimize the Kikuchi Free Energy under some Reducibility Conditions. ~ + + + = 4 5 1 2 3 Kikuchi Free Energy Reducibility Conditions DEX-SMI2008

  27. 5 2 5 1 1 4 5 2 Simplified Cluster Variation Method 2 1 1 4 3 1 3 4 3 DEX-SMI2008

  28. 5 5 5 5 2 2 2 2 4 3 4 3 1 1 1 1 4 3 4 3 Simplified Cluster Variation Method DEX-SMI2008

  29. 5 2 1 5 2 5 2 2 5 4 3 1 1 1 1 1 5 2 4 3 4 3 1 4 3 4 3 Simplified Cluster Variation Method and Factor Graph DEX-SMI2008

  30. Selection of Basic Clusters and their Sub-Clusters in Generalized Belief Propagation on Square Grid The set of Basic Clusters The Set of Basic Clusters and Their Sub-Clusters LBP (Bethe Approx.) DEX-SMI2008

  31. Selection of Basic Clusters and their Sub-Clusters in Generalized Belief Propagation on Square Grid The Set of Basic Clusters and Their Sub-Clusters The set of Basic Clusters GBP (Square Approximation in CVM） DEX-SMI2008

  32. Critical Points for Ising Model (V,E)=Cube Lattice (V,E)=Square Lattice DEX-SMI2008

  33. We reviewed the generalized belief propagation based on cluster variation method. Some extensions of the generalized belief propagation have been given by using the simplified cluster variation method. The relationship between the factor graph and the present extensions by means of the simplified cluster variation method are clarified. Summary DEX-SMI2008

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