CLuP practically achieves $\sim 1.77$ positive and $\sim 0.33$ negative Hopfield model ground state free energy

TL;DR

This paper introduces CLuP algorithms for efficiently approximating ground state energies of positive and negative Hopfield models, achieving near-optimal results with theoretical and practical validation, and revealing intrinsic differences between the models.

Contribution

The paper develops CLuP-based algorithms for Hopfield models and provides theoretical analysis and empirical results demonstrating their effectiveness and insights into model differences.

Findings

CLuP algorithms achieve ground state energy approximations of ~1.77 (positive) and ~0.33 (negative).

Theoretical limits closely match practical performance for small to moderate sizes.

Positive and negative Hopfield models exhibit fundamentally different configuration proximities.

Abstract

We study algorithmic aspects of finding $n$-dimensional \emph{positive} and \emph{negative} Hopfield ($\pm$Hop) model ground state free energies. This corresponds to classical maximization of random positive/negative semi-definite quadratic forms over binary $\left \{\pm \frac{1}{\sqrt{n}} \right \}^n$ vectors. The key algorithmic question is whether these problems can be computationally efficiently approximated within a factor $\approx 1$. Following the introduction and success of \emph{Controlled Loosening-up} (CLuP-SK) algorithms in finding near ground state energies of closely related Sherrington-Kirkpatrick (SK) models [82], we here propose a CLuP$\pm$Hop counterparts for $\pm$Hop models. Fully lifted random duality theory (fl RDT) [78] is utilized to characterize CLuP$\pm$Hop \emph{typical} dynamics. An excellent agreement between practical performance and theoretical predictions is observed. In particular, for $n$ as small as few thousands CLuP$\pm$Hop achieve $\sim 1.77$ and $\sim 0.33$ as the ground state free energies of the positive and negative Hopfield models. At the same time we obtain on the 6th level of lifting (6-spl RDT) corresponding theoretical thermodynamic ($n\rightarrow\infty$) limits $\approx 1.7784$ and $\approx 0.3281$. This positions determining Hopfield models near ground state energies as \emph{typically} easy problems. Moreover, the very same 6th lifting level evaluations allow to uncover a fundamental intrinsic difference between two models: $+$Hop's near optimal configurations are \emph{typically close} to each other whereas the $-$Hop's are \emph{typically far away}.