A Theory of Solving TAP Equations for Ising Models with General Invariant Random Matrices

TL;DR

This paper develops a theoretical framework for solving TAP equations in Ising models with general invariant random matrices, revealing how implicit memory terms can be canceled to simplify the iterative solution process.

Contribution

It introduces a novel analysis of TAP equation iteration using a dynamical functional approach, providing explicit expressions for implicit memory terms in general invariant ensembles.

Findings

Implicit memory terms depend on previous magnetizations.

Cancellation of memory terms simplifies the iteration to Gaussian fields.

Method applied explicitly to random orthogonal ensemble.

Abstract

We consider the problem of solving TAP mean field equations by iteration for Ising model with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields an effective dynamics of a single variable trajectory. Our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles. By subtracting these terms, that depend on magnetizations at previous time steps, the implicit memory terms cancel making the iteration dependent on a Gaussian distributed field only. The TAP magnetizations are stable fixed points if an AT stability criterion is fulfilled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.