Studying Hopfield models via fully lifted random duality theory

Abstract

Relying on a recent progress made in studying bilinearly indexed (bli) random processes in \cite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}, the main foundational principles of fully lifted random duality theory (fl RDT) were established in \cite{Stojnicflrdt23}. We here study famous Hopfield models and show that their statistical behavior can be characterized via the fl RDT. Due to a nestedly lifted nature, the resulting characterizations and, therefore, the whole analytical machinery that produces them, become fully operational only if one can successfully conduct underlying numerical evaluations. After conducting such evaluations for both positive and negative Hopfield models, we observe a remarkably fast convergence of the fl RDT mechanism. Namely, for the so-called square case, the fourth decimal precision is achieved already on the third (second non-trivial) level of lifting (3-sfl RDT) for the positive and on the fourth (third non-trivial) level of lifting (4-sfl RDT) for the corresponding negative model. In particular, we obtain the scaled ground state free energy $\approx 1.7788$ for the positive and $\approx 0.3279$ for the negative model.