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Belief Propagation and Loop Series on Planar Graphs

Belief Propagation and Loop Series on Planar Graphs

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Belief Propagation and Loop Series on Planar Graphs

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  1. Belief Propagation and Loop Series on Planar Graphs Volodya Chernyak, Misha Chertkov and Razvan Teodorescu Department of Chemistry, Wayne State University Theory Division, Los Alamos National Laboratory

  2. Acknowledgements John Klein (Wayne State)

  3. Vertex model formulation for statistical inference problem Traces and graphical traces Gauge-invariant formulation of loop calculus From single-connected partition to dimer matching problem Pfaffian expression for the single partition Tractable models From general binary model to the dimer-monomer matching problem and Pfaffian series on planar graphs Fermion (Grassman), continuous and supersymmetric cases Outline

  4. Statistical inference

  5. Vertex model formulation q-ary variables reside on edges Forney ’01; Loeliger ’01 Baxter Partition function Probability of a configuration Marginal probabilities Reduced variables can be expressed in terms of the derivatives of the free energy with respect to factor-functions

  6. Loop calculus (binary alphabet) Belief Propagation (BP) is exact on a tree

  7. Equivalent models: gauge fixing and transformations Replace the model with an equivalent more convenient model Coordinate approach Invariant approach (i) Introduce an invariant object that describes partition function Z (ii) Different equivalent models correspond to different coordinate choices (gauge fixing) (iii) Gauge transformations are changing the basis sets (i) Introduce a set of gauge transformations that do not change Z (ii) Gauge transformations build new equivalent models General strategy (based on linear algebra) • Replace q-ary alphabet with a q-dimensional vector space • (letters are basis vectors) • (ii) Represent Z by an invariant object graphical trace • (iii) Gauge fixing is a basis set choice • (iv) Gauge transformations are linear transformation of basis sets

  8. Gauge invariance: matrix formulation Gauge transformations of factor-functions with orthogonality conditions do not change the partition function

  9. BP equations: matrix (coordinate) formulation No-loose-ends condition results in BP equations with

  10. BP equations: standard form A standard form of BP equations is reproduced using the following representation for the ground state Side remark: relation to iterative BP

  11. Graphical representation of trace and cyclic trace Trace scalar product Summation over repeating subscripts/superscripts Cyclic trace

  12. Graphic trace and partition function Collection of tensors (poly-vectors) Graphic trace Scalar products Orthogonality condition Tensors and factor-functions

  13. Partition function and graphic trace: gauge invariance Dual basis set of co-vectors (elements of the dual space) Orthogonality condition (two equivalent forms) Graphic trace: Evaluate scalar products (reside on edges) on tensors (reside vertices) Gauge invariance: graphic trace is an invariant object, factor-functions are basis-set dependent “Gauge fixing” is a choice of an orthogonal basis set

  14. Belief propagation gauge and BP equations and excited (painted) states Introduce local ground BP gauge: painted structures with loose ends should be forbidden (in particular, no allowed painted structures in a tree case) or, stated differently, results in BP equations in invariant form:

  15. Loop decomposition: binary case Beliefs (marginal probabilities) A generalized loop visualizes a single-configuration contribution to the partition function in BP gauge

  16. Homogeneous valence-three graphs From arbitrary- to valence-three vertices Transformation for Tanner graphs Planar valence-three graphs dual to a triangulation

  17. Single-connected partition: dimer model Transformation to the extended graph (Fisher’s trick) “Single” partition function (regular loops) Single partition equivalent to dimer-matching problem on the extended graph

  18. Pfaffian expression for dimer model Kasteleyn representation as a Pfaffian (from symmetric to skew-symmetric matrix)

  19. Tractable problems: reduction to single-connected BP equations: loose ends (valence one vertices) forbidden Additional equations: valence three vertices forbidden The model is equivalent to the dimer-matching problem

  20. Pfaffian series for monomer-dimer model Expansion in dimers (triple colored vertices) Each term is computationally tractable

  21. Fermion representation and models Berezin integral Grassman variables Berezin-integral representation of a Pfaffian

  22. Continuous and supersymmetric case: graphical sigma-models Scalar product: the space of states and its dual are equivalent No-loose-end requirement Continuous version of BP equations

  23. Supersymmetric sigma-models: supermanifolds dimension substrate (usual) manifold additional Grassman (anticommuting variables) Functions on a supermanifold Berezin integral (measure in a supermanifold) Any function on a supermanifold can be represented as a sum of its even and odd components

  24. Supersymmetric sigma models: graphic supertrace I Natural assumption: factor-functions are even functions on Introduce parities of the beliefs Follows from the first two BP equations for parities Edge parity is well-defined elements Euler characteristic (number of connected components)

  25. Supersymmetric sigma models: graphic supertrace II Decompose the vector spaces into reduced vector spaces Graphic supertrace decomposition (generalizes the supertrace) results in a multi-reference loop expansion is the graphic trace (partition function) of a reduced model

  26. Summary • We have formulated the statistical inference problem in terms of a graphical trace, which leads to the invariance of the partition function under a set of gauge transformations. • BP equations have been interpreted as a special choice of gauge • The generalized loop (net) expansion appears in a natural way in the BP gauge • For a planar graph we have performed the summation over single-connected loops • A class of tractable models have been identified • A Pfaffian series for a general binary vertex model case on a planar graph have been formulated

  27. Bibliography • M. Chertkov, V.Y. Chernyak, R. Teodorescu • Belief Propagation and Loop Series on Planar Graphs, • arXiv:0802.3950v1 [cond-mat.stat-mech] (2008) • V.Y. Chernyak, M. Chertkov, • Loop Calculus and Belief Propagation for q-ary Alphabet: Loop Tower, • proceeding of ISIT 2007, June2007, Nice, cs.IT/0701086 • M. Chertkov, V.Y. Chernyak, • Loop Calculus Helps to Improve Belief Propagation and • Linear Programming Decodings of Low-Density-Parity-Check Codes, • 44th Allerton Conference (September 27-29, 2006, Allerton, IL); arXiv:cs.IT/0609154 • M. Chertkov, V.Y. Chernyak, • Loop Calculus in Statistical Physics and Information Science, • Phys. Rev. E 73, 065102(R)(2006); cond-mat/0601487 • M. Chertkov, V.Y. Chernyak, • Loop series for discrete statistical models on graphs, • J. Stat. Mech. (2006) P06009,cond-mat/0603189

  28. Path forward: interplay of topological and geometrical equivalence Topological structure: the graph Geometrical structure: factor-functions Use geometrically equivalent models Use topologically equivalent models e.g. Weitz ’06 Combine • + • improving BP • quantum version • etc

  29. Homotopy approach to loop decomposition Bouquet of circles Graph (arbitrary) Equivalent (same homotopy type) by contracting the tree to a point “circles” Both models are equivalent Loop calculus for the bouquet model (independent loops) constitutes a resummation for the original model (generalized loops)

  30. Loop towers for q-ary alphabet: first step A generalized loop defines a vertex model on the corresponding subgraph with (q-1)-ary alphabet (first store above the ground store) q>2 (non-binary case): more than one local excited state Partition function for the subgraph model

  31. Loop-tower expansion for q-ary alphabet Building the next level (store) Loop tower

  32. “Reduced Bethe free energy” (variational approach) Reduced Bethe free energy with is an attempt to approximate the partition function Z in terms of the ground-state contribution in a proper gauge BP equations are recovered by the stationary point conditions Not a standard variational scheme: corrections can be of either sign What is the relation of the introduced functional to the Bethe free energy (Yedidia, Freeman, Weiss ‘01)?

  33. Bethe free energy for q-ary alphabet BP equations can be obtained as stationary points of the Bethe free energy functional of beliefs with natural constraints

  34. Bethe effective Lagrangian Values of beliefs Variation of beliefs

  35. Relation to Bethe free energy Variation of beliefs Variation of the ground state Gauge fixing Reduced Bethe free energy