Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanisms behind this enhancement remain poorly understood. We address this gap by combining a geometric characterization of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, Pinnacle Sharpness, we empirically uncover a rich phase diagram of attractor stability, identifying a Ridge of Optimization where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a Force Antagonism, in which a strong driving force is counterbalanced by a collective feedback force. We theoretically interpret this behavior as a consequence of a specific reorganization of the weight spectrum, which we term Spectral Concentration. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical regime: the leading eigenvalue is amplified to enhance global stability (Direct Force), while the trailing eigenvalues remain finite to sustain high memory capacity (Indirect Force). Together, these results suggest a spectral mechanism by which learning reconciles stability and capacity in high-dimensional associative memory models.

翻译：基于核的学习方法能够显著提升Hopfield网络的存储容量，然而这一增强背后的动力学机制仍不清晰。我们通过将吸引子景观的几何表征与核机器的谱理论相结合，填补了这一空白。利用新提出的度量指标"尖峰锐度"，我们经验性地揭示了吸引子稳定性的丰富相图，并识别出"优化脊"——在此状态下网络在高负载条件下实现最大鲁棒性。从现象学角度看，该脊的特征是"力对抗"：强驱动力与集体反馈力相互制衡。我们从理论上将这种行为解释为权重谱特定重组的结果，并将其命名为"谱集中"。与简单的秩-1塌缩不同，我们的分析表明，优化脊上的网络会自组织进入临界状态：主导特征值被放大以增强全局稳定性（直接力），而尾部特征值保持有限以维持高记忆容量（间接力）。这些结果共同揭示了一种谱机制，通过该机制，学习在高维联想记忆模型中调和了稳定性与容量之间的矛盾。