将霍普菲尔德模型扩展为三维向量形式，提升存储与鲁棒性。
Amorphous Solid Model of Vectorial Hopfield Neural Networks
- 用向量神经元和类非晶固体结构的耦合机制增强记忆能力
- 高连接度下存储容量线性增长，噪声鲁棒性显著提升
- 适合高容量、强抗扰的联想记忆系统设计
我们提出一种三维向量型霍普菲尔德关联记忆模型，其中每个神经元是 $S^2$ 球面上的单位向量，突触耦合由向量型赫布规则生成，形成 $3\times 3$ 块结构。该结构在数学上类似于非晶固体的海森矩阵，诱导出具有深层能量极小值的刚性能量景观，有利于存储模式的稳定。模拟与谱分析表明，该向量网络显著优于经典二进制霍普菲尔德模型：中等连接度下，临界存储比 $γ_c$ 随配位数 $Z$ 近似线性增长；当 $Z \gtrsim 40$ 时进入高连接度区域，$γ_c$ 系统性超过低 $Z$ 时的线性外推值。同时，模式模态与能带间保持持续谱间隙，吸引域扩大，对初始化噪声的鲁棒性增强。因此，几何约束结合类非晶固体结构，使关联记忆在高连接度（$Z \gtrsim 20$-$30$）下表现出更优性能。
原文摘要 · Abstract (English)
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.
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