
Contents
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19.1 Beyond BP: Many states 19.1 Beyond BP: Many states
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19.1.1 Bethe measures 19.1.1 Bethe measures
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19.1.2 A few generic scenarios 19.1.2 A few generic scenarios
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19.2 The 1RSB cavity equations 19.2 The 1RSB cavity equations
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19.2.1 Counting BP fixed points 19.2.1 Counting BP fixed points
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19.2.2 Message passing in the auxiliary model 19.2.2 Message passing in the auxiliary model
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19.2.3 Computing the complexity 19.2.3 Computing the complexity
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19.2.4 Summary 19.2.4 Summary
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19.2.5 Random graphical models and density evolution 19.2.5 Random graphical models and density evolution
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19.2.6 Numerical implementation 19.2.6 Numerical implementation
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19.3 A first application: XORSAT 19.3 A first application: XORSAT
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19.3.1 BP equations 19.3.1 BP equations
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19.3.2 The 1RSB cavity equations 19.3.2 The 1RSB cavity equations
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19.3.3 Complexity 19.3.3 Complexity
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19.4 The special value x = 1 19.4 The special value x = 1
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19.4.1 Back to BP 19.4.1 Back to BP
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19.4.2 A simpler recursion 19.4.2 A simpler recursion
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19.5 Survey propagation 19.5 Survey propagation
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19.5.1 The SP(y) equations 19.5.1 The SP(y) equations
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19.5.2 Energetic complexity 19.5.2 Energetic complexity
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19.5.3 Constraint satisfaction and binary variables 19.5.3 Constraint satisfaction and binary variables
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19.5.4 XORSAT again 19.5.4 XORSAT again
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19.6 The nature of 1RSB phases 19.6 The nature of 1RSB phases
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19.6.1 Dynamical 1RSB 19.6.1 Dynamical 1RSB
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19.6.2 Static 1RSB 19.6.2 Static 1RSB
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19.6.3 When does 1RSB fail? 19.6.3 When does 1RSB fail?
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19.7 Appendix: The SP(y) equations for XORSAT 19.7 Appendix: The SP(y) equations for XORSAT
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Notes Notes
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Cite
Abstract
In graphical models whose factor graph has a locally tree-like structure, belief propagation may fail because variables become correlated at large distances. This phenomenon has been observed in many problems, from satisfiability to colouring or error correcting codes. This chapter describes a physics-based approach for dealing with such a problem, the ‘one step replica symmetry breaking’ (1RSB) cavity method. It is based on the idea of counting solutions to belief propagation equations, and has strong connections with the theory of pure states decomposition. Its algorithmic side, the survey propagation algorithm, is motivated and described in details. The general theory is illustrated through its application to the XORSAT problem studied in Chapter 18.
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