Thouless-Anderson-Palmer equation and self-consistent signal-to-noise analysis for the Hopfield model with three-body interaction

TL;DR

This paper extends the self-consistent signal-to-noise analysis (SCSNA) and TAP equation framework to associative memory neural networks with three-body interactions, providing new insights into their structure and analysis methods.

Contribution

It develops the SCSNA framework for 3-body interaction neural networks and introduces a novel cavity-based method to derive the TAP equation for such systems.

Findings

Derived the TAP equation for 3-body interaction networks.

Connected the Onsager reaction term with the SCSNA output.

Proposed a cavity-based recipe for analyzing complex neural networks.

Abstract

The self-consistent signal-to-noise analysis (SCSNA) is an alternative to the replica method for deriving the set of order parameter equations for associative memory neural network models and is closely related with the Thouless-Anderson-Palmer equation (TAP) approach. In the recent paper by Shiino and Yamana the Onsager reaction term of the TAP equation has been found to be obtained from the SCSNA for Hopfield neural networks with 2-body interaction. We study the TAP equation for an associative memory stochastic analog neural network with 3-body interaction to investigate the structure of the Onsager reaction term, in connection with the term proportional to the output characteristic to the SCSNA. We report the SCSNA framework for analog networks with 3-body interactions as well as a novel recipe based on the cavity concept that involves two cavities and the hybrid use of the SCSNA to obtain the TAP equation.