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.

翻译：基于核逻辑回归的高容量联想记忆模型展现出强大的存储能力，但其稳定性背后的动力学与几何机制仍未被充分理解。本文研究了经KLR训练的霍普菲尔德网络中吸引子盆地的全局几何结构及存储极限的调控机理。我们结合基于随机序列与真实图像嵌入（CIFAR-10）的实证评估、形变实验以及统计信噪比分析。实验表明：对于随机序列，网络存储容量可达$P/N \approx 16$；当结构化数据的有效负载接近$P/N \approx 20$时，仍能保持稳定检索。形变分析显示，“优化脊”上的吸引子由尖锐的相变式边界分隔，其特征为陡峭的有效势垒与临界减速现象。此外，通过将信噪比分析与基于Cover定理的几何参考点进行对比，我们发现实际存储极限的主导因素并非特征空间中的几何可分性缺失，而是对串扰噪声的动力学稳定性丧失。这些结果表明，KLR网络作为高度局部化的基于示例的记忆系统，在接近动力学坍缩临界点处运行，这为设计鲁棒的大规模检索系统提供了新的视角。