对称权重的霍普菲尔德网络可存长期序列，突破传统容量极限。
Beyond Fixed Points: Superpolynomial Capacity of Asymmetric Hopfield Networks
- 用组合数学与数论设计新架构，实现超多项式存储能力。
- n个神经元可存exp(Ω(n/(log n)²))个周期序列，每周期长exp(Ω(√n/log n))。
- 对随机噪声鲁棒性强，适合生物与人工系统中稳定序列记忆。
经典霍普菲尔德网络受限于对称权值，仅能存储静态模式；而异步网络可通过极限环吸引子编码时间序列。然而，传统同步异步网络在存储长序列方面仍面临高容量挑战。本文提出一种简单且稳健的构造方法，在二值神经元与同步更新机制下，使n个神经元支持exp(Ω(n/(log n)²))个不同极限环吸引子，每个周期长度为exp(Ω(√n/log n))，且对随机噪声（翻转概率高达1/2 - o(1)）具有鲁棒性，从而实现序列数量与长度上的超多项式容量。这是首次在异步霍普菲尔德网络中实现此类容量，通过结合组合数学、数论与意见动力学分析获得。研究揭示，同步异步霍普菲尔德网络具备远超以往认知的序列记忆能力，表明在生物与人工神经系统中，仅靠粗粒度结构即可实现鲁棒序列表示，无需复杂非线性机制。
原文摘要 · Abstract (English)
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.
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