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

Abstract

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.

References in corpus (7)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models
• The Mutual Information in Random Linear Estimation
• The mean field Ising model trough interpolating techniques
• Statistical and computational phase transitions in spiked tensor estimation
• The Mutual Information in Random Linear Estimation Beyond i.i.d. Matrices
• Estimation in the spiked Wigner model: A short proof of the replica formula

Cited by in corpus (13)

• Entropy and mutual information in models of deep neural networks
• Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation
• Statistical limits of dictionary learning: random matrix theory and the spectral replica method
• An inference problem in a mismatched setting: a spin-glass model with Mattis interaction
• The multi-species mean-field spin-glass on the Nishimori line
• High-dimensional rank-one nonsymmetric matrix decomposition: the spherical case
• Matrix factorization with neural networks
• Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures
• Gibbs-Bogoliubov inequality on Nishimori line
• Mutual information for low-rank even-order symmetric tensor estimation
• Revisiting the replica trick: Competition between spin glass and conventional order
• Free energy in spin glass models with conventional order
• Toward mean-field bound for critical temperature on Nishimori line