Planar and Surface Graphical Models which are EASY Vladimir Chernyak - PowerPoint PPT Presentation

Introduction Planar is not necessarily easy ... but Surface-Easy Conclusions & Path forward Planar and Surface Graphical Models which are EASY Vladimir Chernyak (1 , 2) and Michael Chertkov 1 1 Center for Nonlinear Studies & Theory Division, LANL 2 Chemistry Department, Wayne State, Detroit “Physics of Algorithms” Workshop Santa Fe, September 1, 2009 http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Planar is not necessarily easy ... but Surface-Easy Conclusions & Path forward Outline 1 Introduction Graphical Models Easy and Difficult Dimer and Ising Models on Planar Graphs 2 Planar is not necessarily easy ... but Holographic Algorithms & Gauge Transformations Edge-Binary models of degree ≤ 3 Edge-Binary Wick Models (of arbitrary degree) 3 Surface-Easy Kasteleyn Conjecture for Dimer Model on Surface Graphs Edge-Binary Graph-Model which are Surface-Easy 4 Conclusions & Path forward Main “take home” message Where do we go from here? http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Binary Graphical Models Forney style - variables on the edges σ ) = Z − 1 � P ( � f a ( � σ a ) f a ≥ 0 a σ ab = σ ba = ± 1 � � Z = f a ( � σ a ) σ 1 = ( σ 12 , σ 14 , σ 18 ) � σ a � �� � � σ 2 = ( σ 12 , σ 23 ) partition function Most Probable Configuration = Maximum Likelihood = Ground State: arg max P ( � σ ) Marginal Probability: e.g. P ( σ ab ) ≡ � σ \ σ ab P ( � σ ) � Partition Function: Z – Our main object of interest Examples http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Easy & Difficult Boolean Problems EASY Any graphical problems on a tree (Bethe-Peierls, Dynamical Programming, BP, TAP and other names) Ground State of a Rand. Field Ferrom. Ising model on any graph Partition function of planar Ising & Dimer models Finding if 2-SAT is satisfiable Decoding over Binary Erasure Channel = XOR-SAT Some network flow problems (max-flow, min-cut, shortest path, etc) Minimal Perfect Matching Problem Some special cases of Integer Programming (TUM) Typical graphical problem, with loops and factor functions of a general position, is DIFFICULT http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Glassy Ising & Dimer Models on a Planar Graph Partition Function of J ij ≷ 0 Ising Model, σ i = ± 1 �� � ( i , j ) ∈ Γ J ij σ i σ j � Z = exp T � σ Partition Function of Dimer Model, π ij = 0 , 1 perfect matching   � � ( z ij ) π ij � � Z = δ π ij , 1  i ∈ Γ j ∈ i π � ( i , j ) ∈ Γ http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Ising & Dimer Classics L. Onsager, Crystal Statistics , Phys.Rev. 65 , 117 (1944) M. Kac, J.C. Ward, A combinatorial solution of the Two-dimensional Ising Model , Phys. Rev. 88 , 1332 (1952) C.A. Hurst and H.S. Green, New Solution of the Ising Problem for a Rectangular Lattice , J.of Chem.Phys. 33 , 1059 (1960) M.E. Fisher, Statistical Mechanics on a Plane Lattice , Phys.Rev 124 , 1664 (1961) P.W. Kasteleyn, The statistics of dimers on a lattice , Physics 27 , 1209 (1961) P.W. Kasteleyn, Dimer Statistics and Phase Transitions , J. Math. Phys. 4 , 287 (1963) M.E. Fisher, On the dimer solution of planar Ising models , J. Math. Phys. 7 , 1776 (1966) F. Barahona, On the computational complexity of Ising spin glass models , J.Phys. A 15 , 3241 (1982) http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Pfaffian solution of the Matching problem 2 2 1 3 1 3 � 4 Detˆ A = Pf[ˆ Z = z 12 z 34 + z 14 z 23 = A ] 4 2   0 − z 12 0 − z 14 1 3 + z 12 0 + z 23 − z 24   ˆ A =   0 − z 23 0 + z 34   4 + z 14 + z 24 − z 34 0 2 Odd-face [Kasteleyn] rule (for signs) 1 3 Direct edges of the graph such that 4 for every internal face the number of Fermion/Grassman Representation edges oriented clockwise is odd http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Graphical Models Planar is not necessarily easy ... but Easy and Difficult Surface-Easy Dimer and Ising Models on Planar Graphs Conclusions & Path forward Planar Spin Glass and Dimer Matching Problems The Pfaffian formula with the “odd-face” orientation rule extends to any planar graph thus proving constructively that Counting weighted number of dimer matchings on a planar graph is easy Calculating partition function of the spin glass Ising model on a planar graph is easy Planar is generally difficult [Barahona ’82] Planar spin-glass problem with magnetic field is difficult Dimer-monomer matching is difficult even in the planar case http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Holographic Algorithms & Gauge Transformations Planar is not necessarily easy ... but Edge-Binary models of degree ≤ 3 Surface-Easy Edge-Binary Wick Models (of arbitrary degree) Conclusions & Path forward Outline 1 Introduction Graphical Models Easy and Difficult Dimer and Ising Models on Planar Graphs 2 Planar is not necessarily easy ... but Holographic Algorithms & Gauge Transformations Edge-Binary models of degree ≤ 3 Edge-Binary Wick Models (of arbitrary degree) 3 Surface-Easy Kasteleyn Conjecture for Dimer Model on Surface Graphs Edge-Binary Graph-Model which are Surface-Easy 4 Conclusions & Path forward Main “take home” message Where do we go from here? http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Holographic Algorithms & Gauge Transformations Planar is not necessarily easy ... but Edge-Binary models of degree ≤ 3 Surface-Easy Edge-Binary Wick Models (of arbitrary degree) Conclusions & Path forward Are there other graphical models which are easy? Holographic Algorithms [Valiant ’02-’08] reduction to dimers via “classical” one-to-one gadgets (e.g. Ising model to dimer model) Ice model to Dimer model ) “holographic” gadgets (e.g. resulted in discovery of variety of new easy planar models Gauge Transformations [Chertkov, Chernyak ’06-’09] Equivalent to the holographic gadgets Gauge Transformations (different gauges = different transformations) Belief Propagation (BP) Loop Calculus/Series is one special choice of the gauge freedom http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Holographic Algorithms & Gauge Transformations Planar is not necessarily easy ... but Edge-Binary models of degree ≤ 3 Surface-Easy Edge-Binary Wick Models (of arbitrary degree) Conclusions & Path forward Are there other graphical models which are easy? Holographic Algorithms [Valiant ’02-’08] reduction to dimers via “classical” one-to-one gadgets (e.g. Ising model to dimer model) Ice model to Dimer model ) “holographic” gadgets (e.g. resulted in discovery of variety of new easy planar models Gauge Transformations [Chertkov, Chernyak ’06-’09] Equivalent to the holographic gadgets Gauge Transformations (different gauges = different transformations) Belief Propagation (BP) Loop Calculus/Series is one special choice of the gauge freedom http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +

Introduction Holographic Algorithms & Gauge Transformations Planar is not necessarily easy ... but Edge-Binary models of degree ≤ 3 Surface-Easy Edge-Binary Wick Models (of arbitrary degree) Conclusions & Path forward Are there other graphical models which are easy? Holographic Algorithms [Valiant ’02-’08] reduction to dimers via “classical” one-to-one gadgets (e.g. Ising model to dimer model) Ice model to Dimer model ) “holographic” gadgets (e.g. resulted in discovery of variety of new easy planar models Gauge Transformations [Chertkov, Chernyak ’06-’09] Equivalent to the holographic gadgets Gauge Transformations (different gauges = different transformations) Belief Propagation (BP) Loop Calculus/Series is one special choice of the gauge freedom http://cnls.lanl.gov/~chertkov/Talks/IT/easy.pdf http://arxiv.org/abs/0902.0320 +