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arXiv 2605.00366cs.NEcs.LG
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核 Hopfield 网络中吸引子边界的几何与动力学分析及存储限制

Geometric and dynamical analysis of attractor boundaries and storage limits in kernel Hopfield networks

Faculty of Information Science, Aichi Institute of Technology（爱知技术大学信息科学学院）

Akira Tamamori

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10 pages, 6 figures
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AI总结：

本文研究了基于核逻辑回归的高容量联想记忆的几何和动力学机制，发现存储容量受动态稳定性影响，而非几何分离性。

摘要：

基于核逻辑回归的高容量联想记忆表现出强大的存储能力，但其稳定性背后的动力学和几何机制尚不明确。本文研究了吸引子盆地的全局几何结构及核 Hopfield 网络的存储限制机制。通过随机序列和现实图像嵌入（CIFAR-10）的实证评估、形态实验和统计信噪比分析，实验表明网络在随机序列上可达到 $P/N \approx 16$ 的存储容量，而在结构数据上在有效负载接近 $P/N \approx 20$ 时保持稳定检索。形态分析显示，优化“顶点”上的吸引子由陡峭的有效势能障碍和临界减速的相变边界分隔。进一步通过比较信噪比分析与受 Cover 定理启发的几何参考点，表明实际存储限制主要由动态稳定性丧失而非特征空间几何分离性不足决定。这些发现表明，KLR 网络作为高度局部化的范例记忆，接近动态崩溃的临界点，为设计鲁棒的大规模检索系统提供了有用的视角。

英文摘要：

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to $P/N \approx 16$, while maintaining stable retrieval for structured data at effective loads near $P/N \approx 20$. Morphing analysis indicates that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries, characterized by steep effective potential barriers and critical slowing down. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.

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