Information-theoretic thresholds from the cavity method

Abstract

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results [Contucci et al.: Communications in Mathematical Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS 2007] about the condensation phase transition in the random graph coloring problem for any number $q\ge 3$ of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes [Montanari: IEEE Transactions on Information Theory 2005].

References in corpus (7)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Hiding Quiet Solutions in Random Constraint Satisfaction Problems
• Mutual Information in Rank-One Matrix Estimation
• The cavity method at zero temperature
• Gibbs Measures and Phase Transitions on Sparse Random Graphs
• The number of solutions for random regular NAE-SAT

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• Machine learning and the physical sciences
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• Typology of phase transitions in Bayesian inference problems
• Storage capacity in symmetric binary perceptrons
• Fundamental limits of low-rank matrix estimation: the non-symmetric case
• Charting the replica symmetric phase
• Information-theoretic and algorithmic thresholds for group testing
• Phase transitions in the $q$-coloring of random hypergraphs
• Spin systems on Bethe lattices
• The random 2-SAT partition function
• Metastability of the Potts ferromagnet on random regular graphs
• The replica symmetric phase of random constraint satisfaction problems
• Phase transitions in the mini-batch size for sparse and dense two-layer neural networks
• Estimating rank-one matrices with mismatched prior and noise: universality and large deviations
• Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures
• An Information-Percolation Bound for Spin Synchronization on General Graphs
• Concentration of multi-overlaps for random ferromagnetic spin models
• Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method
• The rank of random matrices over finite fields
• Noisy group testing via spatial coupling
• Breaking of 1RSB in random MAX-NAE-SAT
• Satisfiability Thresholds for Regular Occupation Problems
• Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems
• Counting colorings of triangle-free graphs
• Frozen $1$-RSB structure of the symmetric Ising perceptron
• Mutual Information in Community Detection with Covariate Information and Correlated Networks
• Inference in Spreading Processes with Neural-Network Priors