Kernel-based learning methods such as Kernel Logistic Regression (KLR) can substantially increase the storage capacity of Hopfield networks, but the principles governing their performance and stability remain largely uncharacterized. This paper presents a comprehensive quantitative analysis of the attractor landscape in KLR-trained networks to establish a solid foundation for their design and application. Through extensive, statistically validated simulations, we address critical questions of generality, scalability, and robustness. Our comparative analysis shows that KLR and Kernel Ridge Regression (KRR) exhibit similarly high storage capacities and clean attractor landscapes under typical operating conditions, suggesting that this behavior is a general property of kernel regression methods, although KRR is computationally much faster. We identify a non-trivial, scale-dependent law for the kernel width $γ$, demonstrating that optimal capacity requires $γ$ to be scaled such that $γN$ increases with network size $N$. This finding implies that larger networks require more localized kernels, in which each pattern's influence is more spatially confined, to mitigate inter-pattern interference. Under this optimized scaling, we provide clear evidence that storage capacity scales linearly with network size~($P \propto N$). Furthermore, our sensitivity analysis shows that performance is remarkably robust with respect to the choice of the regularization parameter $λ$. Collectively, these findings provide a concise set of empirical principles for designing high-capacity and robust associative memories and clarify the mechanisms that enable kernel methods to overcome the classical limitations of Hopfield-type models.

翻译：基于核的学习方法（如核逻辑回归，KLR）可显著提升霍普菲尔德网络的存储容量，但其性能与稳定性的内在原理尚不明确。本文对KLR训练网络的吸引子景观进行系统定量分析，为该类网络的设计与应用奠定坚实基础。通过大规模统计验证的仿真实验，我们探讨了通用性、可扩展性与鲁棒性等关键问题。对比分析表明，在典型运行条件下，KLR与核岭回归（KRR）均展现出相近的高存储容量与清晰吸引子景观，暗示该行为是核回归方法的普遍特性——尽管KRR的计算效率显著更高。我们发现核宽度$γ$存在非平凡且依赖网络规模的缩放规律：最优容量要求$γN$随网络规模$N$增大而增加。这意味着大规模网络需采用更具局部性的核函数（各模式影响范围更受限），以抑制模式间干扰。在此优化缩放下，我们获得明确证据表明存储容量与网络规模呈线性关系（$P \propto N$）。此外，敏感性分析显示，性能对正则化参数$λ$的选取具有惊人鲁棒性。综合这些发现，我们提出一套简洁的经验性设计原则，用于构建高容量、鲁棒的联想记忆模型，并阐明核方法突破传统霍普菲尔德模型局限的核心机制。