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

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 two-body interaction. We study the TAP equation for an associative memory stochastic analog neural network with three-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 on the SCSNA framework for analog networks with three-body interactions as well as provide a recipe based on the cavity concept that involves two cavities and the hybrid use of the SCSNA to obtain the TAP equation.