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)
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 formulation
q-ary variables reside on edges
Forney ’01; Loeliger ‘01
Generalization: continuous/Grassman variables
Probability of a configuration
Marginal probabilities
can be expressed in terms of
the derivatives of the free energy
with respect to factor-functions
Partition function
Reduced variables
Equivalent models: gauge fixing and
transformations
Replace the model with an equivalent more convenient model
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
Coordinate approach
(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)
(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
Gauge invariance: matrix formulation
Gauge transformations of factor-functions
with orthogonality conditions
do not change the partition function
Graphical representation of trace and cyclic trace
Trace
f ij
g ji scalar product
Tr( f )   f j j   fij g ji
Summation over repeating subscripts/superscripts
f in jn
Cyclic trace
g jni1
f i1 j1
g
j1i2
g j2i3
f i2 j2
Tr( f n )   f j1j2 f j2j3 ... f jnj1   fi1 j1 g j1i2 fi2 j2 g j2i3 ... fin jn g jni1
Graphical trace and partition function
Collection of tensors (poly-vectors)
f
a
a1a2 a3
Wab
g
f bb1b2
a 3b1
ab
f  { f a }a 1,..., N  { f aa1 ... an }a 1,... N
a
ia ib
g  {g ab}ab  {g ab
}ab
Wba
Graphic trace
g
ak b j
ab
Tr( f )  Trg ( f a )   (... f aa1 ... ak ... ana ... f bb1 ... b j ... bnb ...)
a
Scalar products
u  w  gab (u  w)
u Wab w Wba
Orthogonality condition
ij
i
i
g ab
 g ab (eab
 ebaj )  eab
 ebaj   ij
Tensors and factor-functions
fa 
n
1
f
(

,...,

)
e

...

e
 a 1 n ab1
abn
1 ...  n
Z  Tr ( f )
Partition function and graphical trace: gauge invariance
Dual basis set of co-vectors
(elements of the dual space)
ab,i (eabj )  i j
 ab  Wab*
Orthogonality condition (two equivalent forms)
ab,i  ba, j   ij
g ab    ab, j  ba , j
j
Graphic trace: Evaluate scalar products (reside on edges) on tensors (reside vertices)
 sb
esb
Tr( f )   f a ( ab ,  ac ,  ad ) f b ( ba ,  bs )...
 
 ab,   ba, 
 ac
ac
e
ab
 ad
ad
e
 ab
eab
ab
 bs
ebs
 ba
eba
Gauge invariance: graphic trace is an invariant object,
factor-functions are basis-set dependent
f a (1,..., n )  ab1 ,1  ...  abn , n ( f a )
“Gauge fixing” is a choice of an orthogonal basis set
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
Gaussian Grassman models: determinants and
Pfaffians
Two key formulas of Gaussain calculus:
det( A)    j d j d j exp

ik
Aikik

Measure is invariant
Pfaff ( A)    j d j exp

ik
Aikik

Requires an ordering choice in the measure
Topological invariants of immersions and SmaleHirsch-Gromov (SHG) Theorem: planar case
Loops in phase space
Immersions
Topologically (homotopy) equivalent
Number of self-intersections
Number of rotations over 2 (spinor structure)
Topological invariants of immersions (II) : planar
case
Generalization to a set of immersions (equality modulo 2)
Number of (self) intersections
Effects of orientations (sign factor)
Spinor structures are the ways to execute a “square root”. In the planar case unique
Topological invariants of immersions: Riemann
surface case
22 g
spinor structures (ways to square root) for a Riemann surface of genus g
Value of the invariant
(quadratic form)
Effects of orientation
Total number of intersections
Edge Binary Wick (EBW) models [VC, Chertkov 09]
Main Theorem [VC, Chertkov 09/surface]
Main “take home” message
Summary
• 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?
Belief propagation gauge and BP equations
Continuous and supersymmetric case:
graphical sigma-models
M
M
M
M
M ab
M ba
Scalar product: the space of states and its dual are equivalent
No-loose-end requirement
Continuous version of BP equations
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
Supersymmetric sigma models: graphic supertrace I
Natural assumption: factor-functions
are even functions on
Introduce parities of the beliefs
pba
pab
BP equations for parities
pab  0,1
Follows from the first two
Edge parity is well-defined
elements
Euler characteristic
(number of connected components)
B1  B0  card ( E )  card (V )
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