Amorphous Solid Model of Vectorial Hopfield Neural Networks

Abstract

We introduce a three-dimensional vectorial extension of the Hopfield associative-memory model in which each neuron is a unit vector on $S^2$ and synaptic couplings are $3\times 3$ blocks generated through a vectorial Hebbian rule. The resulting block-structured operator is mathematically analogous to the Hessian of amorphous solids and induces a rigid energy landscape with deep minima for stored patterns. Simulations and spectral analysis show that the vectorial network substantially outperforms the classical binary Hopfield model. For moderate connectivity, the critical storage ratio $γ_c$ grows approximately linearly with the coordination number $Z$, while for $Z\gtrsim 40$ a high-connectivity regime emerges in which $γ_c$ systematically exceeds the extrapolated low-$Z$ linear fit. At the same time, a persistent spectral gap separates pattern modes from the bulk and basins of attraction enlarge, yielding enhanced robustness to initialization noise. Thus geometric constraints combined with amorphous-solid-inspired structure produce associative memories with superior storage and retrieval performance, especially in the high-connectivity ($Z \gtrsim 20$-$30$) regime.

Figures (7)

• Figure 1: Rendering of a 3D network of $N$ nodes with average coordination number $Z=7$ and nearly uniform orientation distribution.
• Figure 2: Critical storage capacity versus connectivity. Top figure: Dynamical estimate of $\gamma_c$ with standard-deviation error bars across $5$ replicas. Bottom figure: Spectral estimate from the vanishing smallest eigenvalue of the Hessian at each attractor. Both panels show an approximately linear dependence of $\gamma_c$ on $Z$ with closely matching slopes.
• Figure 3: High-connectivity regime of critical storage capacity. Symbols: spectral estimate of $\gamma_c$ (mean $\pm$ sd) versus $Z$. Red line: linear trend $\gamma_c(Z)=0.0053\,(Z-0.166)$ obtained at moderate $Z$ (cf. Fig. \ref{['fig:gamma_two_panels']}). The systematic upward deviation at large $Z$ signals a crossover to an enhanced-capacity regime where $\gamma_c$ exceeds the extrapolated low-$Z$ prediction.
• Figure 4: Critical storage load $\gamma_c = P_c/(N d)$ versus angular standard deviation $\sigma$ of the pattern-orientation distribution, for different connectivities $Z$. Higher connectivity yields larger capacity for all levels of orientational disorder, with $\gamma_c$ increasing with $\sigma$. Error bars: standard deviation across $5$ replicas.
• Figure 5: Monotonic decay of the Lyapunov energy $E(t)$ during synchronous updates for $N=25$, $d=3$, $P=5$. The system converges rapidly to the minimum associated with the retrieved memory.