TAP equations for orthogonally invariant spin glasses at high temperature

TL;DR

This paper proves the validity of TAP equations for orthogonally invariant spin glasses at high temperature, using a novel geometric approach to analyze AMP algorithm convergence in non-i.i.d. models.

Contribution

It establishes the correctness of TAP equations in this setting and introduces a new geometric method for analyzing AMP convergence without i.i.d. assumptions.

Findings

TAP equations hold in high-temperature orthogonally invariant spin glasses.

A new geometric proof of AMP convergence is developed.

Convergence is shown via second moment analysis on a restricted free energy.

Abstract

We study the high-temperature regime of a mean-field spin glass model whose couplings matrix is orthogonally invariant in law. The magnetization of this model is conjectured to satisfy a system of TAP equations, originally derived by Parisi and Potters using a diagrammatic expansion of the Gibbs free energy. We prove that this TAP description is correct in an $L^2$ sense, in a regime of sufficiently high temperature. Our approach develops a novel geometric argument for proving the convergence of an Approximate Message Passing (AMP) algorithm to the magnetization vector, which is applicable in models without i.i.d. couplings. This convergence is shown via a conditional second moment analysis of the free energy restricted to a thin band around the output of the AMP algorithm, in a system of many "orthogonal" replicas.