TAP free energy, spin glasses, and variational inference

2018/08/23 by Zhou Fan, Fan, Zhou, Andrea Montanari +2 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1808.07890

openalex publication_date 2018/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as `Z₂ synchronization'. Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless-Anderson-Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e. the noise level is small enough in synchronization). Namely, we prove that the scaled ℓ₂ distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. Our proof technique is based on upper bounding the expected number of critical points of the TAP free energy using the Kac-Rice formula.

Citations

• Limit laws for Random matrices and free products
• Probabilistic topic models
• Finite Size Corrections and Likelihood Ratio Fluctuations in the Spiked Wigner Model
• An Instability in Variational Inference for Topic Models
• Variational Inference: A Review for Statisticians

Cited by

• Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures
• Variational formula for the logarithmic potential of free additive convolutions

Related

• High temperature TAP upper bound for the free energy of mean field spin glasses
• TAP equations for orthogonally invariant spin glasses at high temperature
• The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings
• TAP variational principle for the constrained overlap multiple spherical Sherrington-Kirkpatrick model
• Phase diagram for the tap energy of the p-spin spherical mean field spin glass model