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

Acknowledgements

John Klein (Wayne State University)John Klein (Wayne State University)Martin Loebl (Charles University)Martin Loebl (Charles University)

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

Forney-style graphical model formulationq-ary variables reside on edgesq-ary variables reside on edges

Probability of a configurationProbability of a configuration Partition functionPartition function

Reduced variablesReduced variablesMarginal probabilitiesMarginal probabilities

can be expressed in terms ofcan be expressed in terms ofthe derivatives of the free energythe derivatives of the free energywith respect to factor-functionswith respect to factor-functions

Forney ’01; Loeliger ‘01Forney ’01; Loeliger ‘01

Generalization: continuous/Grassman variablesGeneralization: continuous/Grassman variables

Equivalent models: gauge fixing and transformations

Replace the model with an equivalent more convenient modelReplace the model with an equivalent more convenient model

Invariant approachInvariant approach Coordinate approachCoordinate approach

(i) Introduce an invariant object that(i) Introduce an invariant object that describes partition function Zdescribes partition function Z(ii) Different equivalent models(ii) Different equivalent models correspond to different coordinate correspond to different coordinate choices (gauge fixing)choices (gauge fixing)(iii) Gauge transformations are(iii) Gauge transformations are changing the basis sets changing the basis sets

(i) Introduce a set of gauge transformations(i) Introduce a set of gauge transformations that do not change Zthat do not change Z(ii) Gauge transformations build new(ii) Gauge transformations build new equivalent modelsequivalent models

General strategy (based on linear algebra)General strategy (based on linear algebra)

(i)(i) Replace q-ary alphabet with a q-dimensional /Replace q-ary alphabet with a q-dimensional /functional functional vector space vector space (ii)(ii) (letters are basis vectors)(letters are basis vectors)(ii) Represent Z by an invariant object (ii) Represent Z by an invariant object graphical tracegraphical trace(iii) Gauge fixing is a basis set choice(iii) Gauge fixing is a basis set choice(iv) Gauge transformations are linear transformation of basis sets(iv) Gauge transformations are linear transformation of basis sets

Gauge invariance: matrix formulation

Gauge transformations of factor-functionsGauge transformations of factor-functions

with orthogonality conditionswith orthogonality conditions

do not change the partition functiondo not change the partition function

Graphical representation of trace and cyclic trace

jiij

jj gfffTr )(

132

22

21

11

13

2

2

1......)( ij

jiij

jiij

jijj

jj

jj

n n

nnngfgfgfffffTr

ijf jig

21ijg

nn jif

11 jif

22 jif

1ijng32ijg

TraceTrace

Cyclic traceCyclic trace

Summation over repeating subscripts/superscriptsSummation over repeating subscripts/superscripts

scalar productscalar product

Graphical trace and partition function

bbbf 2113ba

abgaaaaf 321

abW baW

NaaaaNa

a

anfff ,...1...,...,1 }{}{1

baiiabbaabbaggg }{}{

...)...(...)()( ............ 11

bbbb

aaaa

a

ag

bnjankfffTrfTr

jkbaabg

)( wugwu ab

abWu baWwijj

baiab

jba

iabab

ijab eeeegg )(

n

n

nababnaa eeff

...

1

1

1

1...),...,(

Collection of tensors (poly-vectors)Collection of tensors (poly-vectors)

Scalar productsScalar products

Graphic traceGraphic trace

Orthogonality conditionOrthogonality condition

Tensors and factor-functionsTensors and factor-functions )( fTrZ

Partition function and graphical trace: gauge invariance

ji

jabiab e )(,

*abab W ijjbaiab ,,

jjbajababg ,,

Dual basis set of co-vectorsDual basis set of co-vectors(elements of the dual space)(elements of the dual space) Orthogonality condition (two equivalent forms)Orthogonality condition (two equivalent forms)

)(...),...,( ,,1 11

aababna ff

nn

abab baab ,,

ababe ba

bae

adade

acace

sbsbe

bsbse

)...,(),,()( bsbabadacaba fffTr

Graphic trace: Evaluate scalar products (reside on edges) on tensors (reside vertices)Graphic trace: Evaluate scalar products (reside on edges) on tensors (reside vertices)

Gauge invariance: graphic trace is an invariant object,Gauge invariance: graphic trace is an invariant object,factor-functions are basis-set dependentfactor-functions are basis-set dependent

““Gauge fixing” is a choice of an orthogonal basis setGauge fixing” is a choice of an orthogonal basis set

Supersymmetric sigma-models: calculus of Grassman variables and supermanifolds

dimensiondimension

substrate (usual) manifoldsubstrate (usual) manifold

additional Grassman (anticommuting variables)additional Grassman (anticommuting variables)

Functions on a supermanifoldFunctions on a supermanifold

Berezin integral (measure in a supermanifold)Berezin integral (measure in a supermanifold)

Any function on a supermanifold can be representedAny function on a supermanifold can be representedas a sum of its even and odd componentsas a sum of its even and odd components

Gaussian Grassman models: determinants and Pfaffians

Two key formulas of Gaussain calculus:Two key formulas of Gaussain calculus:

j ik kiikjj AddA exp)det(

j ik kiikj AdAPfaff exp)(

Measure is invariantMeasure is invariant

Requires an ordering choice in the measureRequires an ordering choice in the measure

Topological invariants of immersions and Smale-Hirsch-Gromov (SHG) Theorem: planar case

ImmersionsImmersions Loops in phase spaceLoops in phase space

Topologically (homotopy) equivalentTopologically (homotopy) equivalent

Number of self-intersectionsNumber of self-intersections Number of rotations over 2 (spinor structure)Number of rotations over 2 (spinor structure)

Topological invariants of immersions (II) : planar case

Generalization to a set of immersions (equality modulo 2)Generalization to a set of immersions (equality modulo 2)

Effects of orientations (sign factor)Effects of orientations (sign factor)Number of (self) intersectionsNumber of (self) intersections

Spinor structures are the ways to execute a “square root”. In the planar case uniqueSpinor structures are the ways to execute a “square root”. In the planar case unique

Topological invariants of immersions: Riemann surface case

spinor structures (ways to square root) for a Riemann surface of genus spinor structures (ways to square root) for a Riemann surface of genus g g g22

Effects of orientationEffects of orientationValue of the invariantValue of the invariant(quadratic form)(quadratic form) Total number of intersectionsTotal number of intersections

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

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

Main “take home” message

• 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

Belief propagation gauge and BP equations

Continuous and supersymmetric case: graphical sigma-models

abM baMM

M

MM

Scalar product: the space of states and its dual are equivalentScalar product: the space of states and its dual are equivalent

No-loose-end requirementNo-loose-end requirement

Continuous version of BP equationsContinuous version of BP equations

Supersymmetric sigma-models: supermanifolds

dimensiondimension

substrate (usual) manifoldsubstrate (usual) manifold

additional Grassman (anticommuting variables)additional Grassman (anticommuting variables)

Functions on a supermanifoldFunctions on a supermanifold

Berezin integral (measure in a supermanifold)Berezin integral (measure in a supermanifold)

Any function on a supermanifold can be representedAny function on a supermanifold can be representedas a sum of its even and odd componentsas a sum of its even and odd components

Supersymmetric sigma models: graphic supertrace I

1,0abpabp bap

)()(01 VcardEcardBB

Natural assumption: factor-functionsNatural assumption: factor-functionsare even functions onare even functions on

Introduce parities of the beliefsIntroduce parities of the beliefs

BP equations for paritiesBP equations for parities Follows from the first twoFollows from the first two

Edge parity is well-definedEdge parity is well-definedelementselements

(number of connected components)(number of connected components)

Euler characteristicEuler characteristic

Supersymmetric sigma models: graphic supertrace II

Decompose the vector spacesDecompose the vector spaces

Graphic supertrace decomposition (generalizes the supertrace)Graphic supertrace decomposition (generalizes the supertrace)

results in a multi-reference loop expansionresults in a multi-reference loop expansion

into reduced vector spacesinto reduced vector spaces

is the graphic trace (partition function) of a reduced modelis the graphic trace (partition function) of a reduced model