Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

Korada, Satish Babu; Macris, Nicolas

doi:10.1007/s10955-009-9781-6

Cited by 48 publications

(46 citation statements)

References 35 publications

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“…Strictly speaking the analysis [11] does not cover the widest possible regime of dense graphs (see section two for details). We note that the mutual information of rank-one matrix factorization had also been determined earlier in [13] for the symmetric case and more recently for the general case in [14,15] using a spatial coupling method.…”

Section: Introductionsupporting

confidence: 53%

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

Barbier

Chan

Macris

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We rigorously derive a single-letter variational expression for the mutual information of the asymmetric two-groups stochastic block model in the dense graph regime. Existing proofs in the literature are indirect, as they involve mapping the model to a rank-one matrix estimation problem whose mutual information is then determined by a combination of methods (e.g., interpolation, cavity, algorithmic, spatial coupling). In this contribution we provide a self-contained and direct proof using only the recently introduced adaptive interpolation method.

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Section: Introductionsupporting

confidence: 53%

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

Barbier

Chan

Macris

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…We will prove the following theorem, already proved using the adaptive interpolation method in [20] (formulated in a more technical discrete time setting). Note that the theorem was also already proved in [34] for a binary Bernoulli signal and also more recently using different (and more involved) techniques in [12,18,35].…”

Section: Rank-one Matrix Estimation or Wigner Spike Modelmentioning

confidence: 81%

The adaptive interpolation method for proving replica formulas. Applications to the Curie–Weiss and Wigner spike models

Barbier¹,

Macris²

2019

J. Phys. A: Math. Theor.

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In this contribution we give a pedagogic introduction to the newly introduced adaptive interpolation method to prove in a simple and unified way replica formulas for Bayesian optimal inference problems. Many aspects of this method can already be explained at the level of the simple Curie-Weiss spin system. This provides a new method of solution for this model which does not appear to be known. We then generalize this analysis to a paradigmatic inference problem, namely rank-one matrix estimation, also refered to as the Wigner spike model in statistics. We give many pointers to the recent literature where the method has been succesfully applied.Keywords adaptive interpolation · Bayesian inference · replica formula · matrix estimation · Wigner spike model · Curie-Weiss model · spin systems 1 IntroductionThe replica method from statistical mechanics has been applied to Bayesian inference problems (e.g., coding, estimation) already two decades ago [1,2]. Rigorous proofs of the formulas for the mutual informations/entropies/free energies stemming from this method, have for a long time only been partial, consisting generally of one sided bounds [3,4,5,6,7,8,9,10]. It is only quite recently that there has been a surge of progress using various methods -namely spatial coupling [11,12,13,14,15], information theory [16], and rigorous versions of the cavity method [17,18,19]-to derive full proofs, but which are typically quite complicated. Recently we introduced [20] a powerful evolution of the Guerra-Toninelli [21,22,23] interpolation method -called adaptive interpolation-that allows to fully prove the replica formulas in a quite simple and unified way for Bayesian inference problems. In this contribution we give a pedagogic introduction to this new method.We first illustrate how the adaptive interpolation method allows to solve the well known Curie-Weiss model. For this model a simple version of the method contains most of the crucial ingredients. The present rigorous method of solution is new and perhaps simpler

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“…The proof we shall present is a straightforward generalization of the one presented in [10] for the pure tensor case, and in [37] for the matrix case, and it is based on two main ingredients. The first one is the Guerra interpolation method applied on the Nishimori line [37,54,55], in which we construct an interpolating Hamiltonian that depends on a parameter t ∈ [0; 1] that is used to move from the original Hamiltonian of Eq. (A6), to the one corresponding to a scalar denoising problem whose free entropy is given by the first term in Eq.…”

Section: Approximate Message Passing and Bethe Free Entropymentioning

confidence: 99%

Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference

et al. 2020

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Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work we perform an analytic study of the performances of the one most commonly considered in physics, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked matrix-tensor model. The typical behaviour of this algorithm is described by a system of integro-differential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is sub-optimal with respect to the one given by AMP. This phenomenon is due to the residual glassiness present in that region of parameters. We present also a simple heuristic expression of the transition line which appears to be in agreement with the numerical results. CONTENTS

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Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

Cited by 48 publications

References 35 publications

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

The adaptive interpolation method for proving replica formulas. Applications to the Curie–Weiss and Wigner spike models

Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference

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