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9780198570837

Information, Physics, and Computation

by ;
  • ISBN13:

    9780198570837

  • ISBN10:

    019857083X

  • Format: Hardcover
  • Copyright: 2009-03-27
  • Publisher: Oxford University Press
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Summary

This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. It is accessible to graduate students and researchers without a specific training in any of these fields. The selected topics include spin glasses, error correcting codes, satisfiability, and are central to each field. The approach focuses on large random instances, adopting a common probabilistic formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to study phase transitions.

Author Biography


Professor Marc Mezard
CNRS Research Director at Universite de Paris Sud and Professor at Ecole Polytechnique, France
Marc Mezard received his PhD in 1984. He was hired in CNRS in 1981 and became research director in 1990 at Ecole Normale Superieure. He joined the Universite Paris Sud in 2001. He spent extensive periods in Rome University, in the KITP (Santa Barbara) and in MSRI (Berkeley). Author of about 150 publications, he has been
awarded the silver medal of CNRS in 1990 and the Ampere price of the French academy of science in 1996. Dr Andrea Montanari
Assistant Professor, Stanford University and CNRS France
Andrea Montanari received a Laurea degree in Physics in 1997, and a Ph. D. in Theoretical Physics in 2001 (both from Scuola Normale Superiore in Pisa, Italy). He has been post-doctoral fellow at Laboratoire de Physique Theorique de l'Ecole Normale Superieure (LPTENS), Paris, France, and the Mathematical Sciences Research Institute, Berkeley, USA. Since 2002 he is Charge de Recherche (a permanent research position with Centre National de la Recherche Scientifique, CNRS) at LPTENS.
In September 2006 he joined Stanford University as Assistant Professor in the Departments of Electrical Engineering and Statistics.
In 2006 he was awarded the CNRS bronze medal for theoretical physics.

Table of Contents

Independence
The random energy modelp. 93
Definition of the modelp. 93
Thermodynamics of the REMp. 94
The condensation phenomenonp. 100
A comment on quenched and annealed averagesp. 101
The random subcube modelp. 103
Notesp. 105
The random code ensemblep. 107
Code ensemblesp. 107
The geometry of the random code ensemblep. 110
Communicating over a binary symmetric channelp. 112
Error-free communication with random codesp. 120
Geometry again: Sphere packingp. 123
Other random codesp. 126
A remark on coding theory and disordered systemsp. 127
Appendix: Proof of Lemma 6.2p. 128
Notesp. 128
Number partitioningp. 131
A fair distribution into two groups?p. 131
Algorithmic issuesp. 132
Partition of a random list: Experimentsp. 133
The random cost modelp. 136
Partition of a random list: Rigorous resultsp. 140
Notesp. 143
Introduction to replica theoryp. 145
Replica solution of the random energy modelp. 145
The fully connected p-spin glass modelp. 155
Extreme value statistics and the REMp. 163
Appendix: Stability of the RS saddle pointp. 166
Notesp. 169
Models on Graphs
Factor graphs and graph ensemblesp. 173
Factor graphsp. 173
Ensembles of factor graphs: Definitionsp. 180
Random factor graphs: Basic propertiesp. 182
Random factor graphs: The giant componentp. 187
The locally tree-like structure of random graphsp. 191
Notesp. 194
Satisfiabilityp. 197
The satisfiability problemp. 197
Algorithmsp. 199
Random K-satisfiability ensemblesp. 206
Random 2-SATp. 209
The phase transition in random K(>q; 3)-SATp. 209
Notesp. 217
Low-density parity-check codesp. 219
Definitionsp. 220
The geometry of the codebookp. 222
LDPC codes for the binary symmetric channelp. 231
A simple decoder: Bit flippingp. 236
Notesp. 239
Spin glassesp. 241
Spin glasses and factor graphsp. 241
Spin glasses: Constraints and frustrationp. 245
What is a glass phase?p. 250
An example: The phase diagram of the SK modelp. 262
Notesp. 265
Bridges: Inference and the Monte Carlo methodp. 267
Statistical inferencep. 268
The Monte Carlo method: Inference via samplingp. 272
Free-energy barriersp. 281
Notesp. 287
Short-Range Correlations
Belief propagationp. 291
Two examplesp. 292
Belief propagation on tree graphsp. 296
Optimization: Max-product and min-sump. 305
Loopy BPp. 310
General message-passing algorithmsp. 316
Probabilistic analysisp. 317
Notesp. 325
Decoding with belief propagationp. 327
BP decoding: The algorithmp. 327
Analysis: Density evoluationp. 329
BP decoding for an erasure channelp. 342
The Bethe free energy and MAP decodingp. 347
Notesp. 352
The assignment problemp. 355
The assignment problem and random assignment ensemblesp. 356
Message passing and its probabilistic analysisp. 357
A polynomial message-passing algorithmp. 366
Combinatorial resultsp. 371
An exercise: Multi-index assignmentp. 376
Notesp. 378
Ising models on random graphsp. 381
The BP equations for Ising spinsp. 381
RS cavity analysisp. 384
Ferromagnetic modelp. 386
Spin glass modelsp. 391
Notesp. 399
Long-Range Correlations
Linear equations with Boolean variablesp. 403
Definitions and general remarksp. 404
Belief propagationp. 409
Core percolation and BPp. 412
The Sat-Unsat threshold in random Xorsatp. 415
The Hard-Sat phase: Clusters of solutionsp. 421
An alternative approach: The cavity methodp. 422
Notesp. 427
The 1RSB cavity methodp. 429
Beyond BP: Many statesp. 430
The 1RSB cavity equationsp. 434
A first application: Xorsatp. 444
The special value x=1p. 449
Survey propagationp. 453
The nature of 1RSB phasesp. 459
Appendix: The SP(y) equations for Xorsatp. 463
Notesp. 465
Random K-satisfiabilityp. 467
Belief propagation and the replica-symmetric analysisp. 468
Survey propagation and the 1RSB phasep. 474
Some ideas about the full phase diagramp. 485
An exercise: Colouring random graphsp. 488
Notesp. 491
Glassy states in coding theoryp. 493
Local search algorithms and metastable statesp. 493
The binary erasure channelp. 500
General binary memoryless symmetric channelsp. 506
Metastable states and near-codewordsp. 513
Notesp. 515
An ongoing storyp. 517
Gibbs measures and long-range correlationsp. 518
Higher levels of replica symmetry breakingp. 524
Phase structure and the behaviour of algorithmsp. 535
Notesp. 538
Symbols and notationp. 541
Equivalence relationsp. 541
Orders of growthp. 542
Combinatorics and probabilityp. 543
Summary of mathematical notationp. 544
Information theoryp. 545
Factor graphsp. 545
Cavity and message-passing methodsp. 545
Referencesp. 547
Indexp. 565
Table of Contents provided by Ingram. All Rights Reserved.

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