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

arXiv: https://arxiv.org/abs/2605.14059 · HTML: https://mlsanity.com/paper/2605.14059/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 10 of 10)

- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models](https://mlsanity.com/paper/2202.04557/) (2022)
- [Supervised Hebbian Learning](https://mlsanity.com/paper/2203.01304/) (2022)
- [Dreaming neural networks: forgetting spurious memories and reinforcing pure ones](https://mlsanity.com/paper/1810.12217/) (2018)
- [Dreaming neural networks: rigorous results](https://mlsanity.com/paper/1812.09077/) (2018)
- [Thermodynamics of bidirectional associative memories](https://mlsanity.com/paper/2211.09694/) (2022)
- [Regularization, early-stopping and dreaming: a Hopfield-like setup to address generalization and overfitting](https://mlsanity.com/paper/2308.01421/) (2023)
- [Supervised and Unsupervised protocols for hetero-associative neural networks](https://mlsanity.com/paper/2505.18796/) (2025)

## Built on by (most cited first, 1 of 1)

- [Exponential Capacity in Multilayer Hetero-Associative Neural Networks](https://mlsanity.com/paper/2607.29554/) (2026)

## Abstract (arXiv)

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 αc(N)=αc(∞)-c,N-1/2, extrapolating to an asymptotic operational limit αc(∞)≈ 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.