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. Outline 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

4. Statistical inference

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

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 Invariant approach Coordinate approach (i) Introduce an invariant object that describes partition function Z describes partition function Z (ii) Different equivalent models correspond to different coordinate correspond to different coordinate choices (gauge fixing) choices (gauge fixing) (iii) Gauge transformations are changing the basis sets changing the basis sets (i) Introduce a set of gauge transformations that do not change Z that do not change Z (ii) Gauge transformations build new equivalent models equivalent models General strategy (based on linear algebra) (i)Replace q-ary alphabet with a q-dimensional vector space (ii)(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 Cyclic trace Summation over repeating subscripts/superscripts scalar product

12. Graphic trace and partition function Collection of tensors (poly-vectors) Scalar products Graphic trace 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 Introduce local ground and excited (painted) states 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 A generalized loop visualizes a single-configuration contribution to the partition function in BP gauge Beliefs (marginal probabilities)

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 equivalent to dimer-matching problem on the extended graph “Single” partition function (regular loops)

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 Grassman variables Berezin integral 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 BP equations for parities Follows from the first two Edge parity is well-defined elements (number of connected components) Euler characteristic

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

26. 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 Summary

27. Bibliography M. Chertkov, V.Y. Chernyak, R. Teodorescu 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, 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, 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, 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, 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 topologically equivalent models Use geometrically equivalent models Combine e.g. Weitz ’06 + improving BP improving BP quantum version quantum version etc etc

29. Homotopy approach to loop decomposition Graph (arbitrary) Bouquet of circles 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) Partition function for the subgraph model q>2 (non-binary case): more than one local excited state

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 is an attempt to approximate the partition function Z in terms of the ground-state contribution in a proper gauge with 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 Variation of beliefs Values of beliefs

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