Beyond Fixed Points: Superpolynomial Capacity of Asymmetric Hopfield Networks

2026-05-23

AI Summary

This work addresses the limitations of traditional symmetric Hopfield networks, which can only store static patterns, and existing heterogeneous or asynchronous models, which struggle with high-capacity long-sequence memory. The authors propose a novel mechanism based on limit-cycle attractors within the classical framework of synchronous, binary-neuron asymmetric Hopfield networks, achieving—for the first time—super-polynomial storage capacity in both the number and length of sequences. By integrating combinatorics, number theory, and opinion dynamics analysis, they construct a temporally structured memory architecture exhibiting high robustness to noise: with merely $n$ neurons, the network supports $\exp(\Omega(n/(\log n)^2))$ distinct limit cycles, each of period up to $\exp(\Omega(\sqrt{n}/\log n))$, and remains stable even under bit-flip noise probabilities approaching $1/2$.

Abstract

Classical Hopfield networks are limited to static patterns due to symmetric weights, whereas asymmetric networks can encode temporal sequences via limit-cycle attractors. Achieving high-capacity storage of long sequences in classical synchronous asymmetric networks, however, has remained a challenge. We present a simple and robust construction within the classical asymmetric Hopfield model with binary neurons and synchronous updates, that allows $n$ neurons to support $\exp\!\big(Ω(n/(\log n)^2)\big)$ distinct limit-cycle attractors, each with period $\exp\!\big(Ω(\sqrt n/\log n)\big)$ and robust to random noise with flip probability up to $\frac12-o(1)$, yielding superpolynomial capacity in both the number and length of stored sequences. This is the first demonstration of such capacity for asymmetric Hopfield networks, which we obtain by combining results from combinatorics, number theory and the analysis of opinion dynamics. Our findings show that synchronous asymmetric Hopfield networks possess a sequence-memory capacity which is larger and more robust than previously recognized, demonstrating that, in both biological and artificial neural systems, robust sequence representation can be achieved through coarse architectural motifs rather than complex nonlinearities.

Problem

• asymmetric Hopfield networks
• sequence storage
• limit-cycle attractors
• memory capacity
• temporal sequences

Innovation

• asymmetric Hopfield networks
• limit-cycle attractors
• superpolynomial capacity
• sequence memory
• robustness to noise