自旋玻璃与消息传递算法Spin Glass and Message-Passing 概要

New vertex partially covered
New vertex always covered
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Mean-field theory result is lower than experimental values for c > e=2.7183
2.7183
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假定的相空间结构
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distribution of equilibrium free-energies (contl.)
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Which thermodynamic states contribute to the equilibrium properties? • If • If
Excited macrostates matter!
thevertexcoverproblem38alwaysuncoveredalwayscoveredunfrozenweigthartmannprl2000pre200139newvertexuncoverednewvertexpartiallycoverednewvertexalwayscovered40meanfieldtheoryresultislowerthanexperimentalvaluesforce271832718341假定的相空间结构42引入参数y43neighborsvertexiprobabilityvcsizeincreasereweightedprobabilityallunfrozenoralwayscoveredalwaysuncovered0atleastonealwaysuncoveredunfrozenoralwayscovered1444546同样的消息传递的算法可以用于解决神经网络?信息系统?满足性问题?
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部分参考文献
1. Mezard, Parisi, Virasoro, “Spin Glass Theory and Beyond” (World Scientific, 1987） Mezard, Parisi, “The Bethe lattice spin glass revisited”, European Physics Journal B 20: 217-233 (2001) Mezard, Parisi, Zecchina, “Analytic and algorithmic solution of random satisfiability problems”, Science 297: 812-815 (2002)
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There are many optimal solutions for a given graph
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Three types of vertices：
(1) vertices that are always covered (frozen vertices, )
(2) vertices that are always uncovered (frozen vertices, )
……
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Equilibrium (static) and dynamical properties of complex systems are both difficult and interesting • Quenched randomness, frustration, nonself averaging, …, broken ergodicity. • NP-complete combinatorial optimizations, message-passing algorithms for information science (CDMA, for example!), ystems.
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自旋玻璃理论： 自由能分
Statistical mechanics of a (simple) system in equilibrium is well-established. Partition function, free energy, …. Mean-field treatment. Phase transitions. Correlation length, scaling exponents, … . Renormalization flow.
引入参数 y
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neighbors all unfrozen or always covered at least one always uncovered
vertex i always uncovered unfrozen or always covered
probability
VC size re-weighted increase probability
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How to calculate the grand free energy? The cavity approach
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N
N+2
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Population Dynamics Simulation
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Message-Passing Algorithms
3-SAT 问题
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顶点覆盖问题
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自旋玻璃：无序与阻错系统的简单模型
• 3D regular lattice
(Edwards-Anderson, 1975)
• Complete graph
(Sherrington-Kirkpatrick 1975)
• Random Poisson graph
(Viana-Bray, 1985)
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ergodic vs non-ergodic
?
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repeated heating—annealing and the equilibrium Gibbs measure
Complexity （复杂度）
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distribution of equilibrium free-energies
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This graph is covered, but not optimally covered.
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Minimal Vertex Cover Problem
• A vertex cover of the global minimal size.
• Is a NP-hard optimization problem. • Efficient algorithms for constructing nearoptimal solutions for a given graph.
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Macrostates of minimal free energy density matter!
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3-spin-Interaction Ising model on a complete graph
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the mean free energy density
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Overlap Distribution
自旋玻璃与消息传递算法
Spin Glass and Message-Passing Algorithms
周海军 /~zhouhj/ 中国科学院理论物理研究所
提纲
1。自旋玻璃理论
• 基本图像与平衡自 由能分布 • 空腔方法
2。消息传递算法
• Vertex-Cover 问题, 3SAT 问题 • Survey Propagation算 法
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…, but non-equilibrium dynamics of even a simple system may be difficult to understand
Formal framework. Connection with equilibrium.
Glassy dynamics. Why relaxation becomes so low and non-exponential?
(3) vertices that are covered in some solutions and uncovered in the remaining solutions (unfrozen vertices, )
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Mean-field analysis of the minimal vertex cover problem on a random graph
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The vertex cover problem
自洽的空穴场方法
覆盖还是不覆盖?
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Weigt, Hartmann, PRL (2000), PRE (2001)
always uncovered always covered =
unfrozen
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New vertex un-covered
What we learned from an equilibrium statistical mechanics course?
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What we learned from an equilibrium statistical mechanics course? (contl.)
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What we learned from an equilibrium statistical mechanics course? (contl.)
0 +1
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同样的消息传递的算法可以用于解决 神经网络，信息系统，满足性问题， …， 中的许多计算困难
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Program and School in Beijing 2008
• ICTP-ITP Spring School on “Statistical Physics and Interdisciplinary Applications” March 03-14, 2008 • KITPC Program “Collective Dynamics in Information Systems” March 01-April 15, 2008
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自旋玻璃理论： 空腔方法（cavity method)
Let’s define an artificial system!
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Some examples of the grand free energy： 2-body interactions
Beta=1.25 Beta=+infinity The max-2-SAT problem The +/- J spin-glass model on a random regular graph of degree K=6