1. Planar and surface graphical models which are easy Vladimir Chernyak and Michael Chertkov Department of Chemistry, Wayne State University Department of Mathematics, Wayne State University Theory Division, Los Alamos National Laboratory

2. Acknowledgements John Klein (Wayne State University) Martin Loebl (Charles University)

3. Outline Forney-style graphical models: generically hard A family of graphical models that turn out to be easy Gauge invariance and linear algebra: traces and graphical traces Calculus of Grassman (anticommuting) variables: determinants and Pfaffians Topological invariants of immersions: equivalence between certain binary and Gaussain Grassman (fermion) models Planar versus surface easy

4. Forney-style graphical 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 Generalization: continuous/Grassman variables

8. 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 /functional 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

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

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

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

12. Partition function and graphical 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

13. Supersymmetric sigma-models: calculus of Grassman variables and 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

14. Gaussian Grassman models: determinants and Pfaffians Two key formulas of Gaussain calculus: Measure is invariant Requires an ordering choice in the measure

15. Topological invariants of immersions and Smale- Hirsch-Gromov (SHG) Theorem: planar case Immersions Loops in phase space Topologically (homotopy) equivalent Number of self-intersections Number of rotations over 2 (spinor structure)

16. Topological invariants of immersions (II) : planar case Generalization to a set of immersions (equality modulo 2) Effects of orientations (sign factor) Number of (self) intersections Spinor structures are the ways to execute a “square root”. In the planar case unique

17. Topological invariants of immersions: Riemann surface case spinor structures (ways to square root) for a Riemann surface of genus g Effects of orientation Value of the invariant (quadratic form) Total number of intersections

18. Edge Binary Wick (EBW) models [VC, Chertkov 09]

24. Main Theorem [VC, Chertkov 09/surface]

26. Main “take home” message

27. We have identified a family of graphical models that are planar/surface easy We have established two ingredients that make these models easy Ingredient #1: Invariant nature of linear algebra(traces and graphical traces) Ingredient #2: Topological invariants of immersions (intersections, self-intersections and spinor structures) Question (and maybe path forward): Can the easy family be extended? Summary

28. Belief propagation gauge and BP equations

29. 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

30. 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

31. 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

32. 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