Finite-size scaling of hetero-associative retrieval in continuous-signal-driven Ising spin systems

TL;DR

This paper introduces a multilayer Ising model framework that bridges continuous real-world signals and discrete associative memory, analyzing finite-size effects and demonstrating cross-modal recall in sleep data.

Contribution

It develops a novel continuous-to-Ising encoding scheme combined with pseudo-inverse couplings, revealing finite-size scaling and operational limits in hetero-associative retrieval.

Findings

Operational capacity approaches 0.50 in the thermodynamic limit.

Parallel updates trigger signal avalanches, sequential updates resolve superpositions.

The model successfully reconstructs sleep states from noisy EEG and EOG signals.

Abstract

Real-world physical signals are continuous and high-dimensional, yet the statistical-mechanics machinery of associative memory operates on discrete Ising spins. We bridge this divide through a multilayer Ising framework that couples a geometry-preserving continuous-to-Ising encoder (PCA whitening composed with SimHash random-hyperplane projection) to Kanter-Sompolinsky pseudo-inverse memory couplings, embedded directly into the local-field equations of a tri-layer hetero-associative system. The pseudo-inverse correction renders the equal-weight mixture state thermodynamically unstable, so that thermal fluctuations break the cross-modal symmetry and select a single global winner. We further establish a dynamical duality: parallel (Little) updates are structurally required to ignite the cross-modal signal avalanche from a single cued layer, whereas sequential (Glauber) sweeps resolve symmetric superpositions. The operational storage capacity obeys the Amit-Gutfreund-Sompolinsky finite-size correction $\alpha_c(N)=\alpha_c(\infty)-c\,N^{-1/2}$, extrapolating to an asymptotic operational limit $\alpha_c(\infty)\approx 0.50$ under macroscopic-basin retrieval. Applied to multi-channel sleep polysomnography (PhysioNet Sleep-EDF), the architecture reconstructs the macroscopic sleep state on parietal EEG and EOG axes from a single noisy frontal-EEG cue, demonstrating cross-modal recall in the presence of quenched biological disorder.