揭示高容量核霍普菲尔德网络的稳定边界与信息几何的深层联系。
Spectral Concentration at the Edge of Stability: Information Geometry of Kernel Associative Memory
- 在统计流形上分析网络动态，发现稳定边界对应信息矩阵奇异点。
- 极端稳定性源于黎曼空间中的双重平衡机制，非传统欧氏力对抗。
- 统一学习动态与存储容量，为自组织临界性提供几何解释。
高容量核霍普菲尔德网络表现出一种由极端稳定性定义的优化脊（Ridge of Optimization）。尽管此前与谱集中现象相关联，其起源仍不明确。本文在统计流形上分析网络动力学，揭示该优化脊对应于稳定性边缘——即费雪信息矩阵变为奇异的临界边界。我们证明，看似对立的欧氏力效应实为黎曼空间中双重平衡的表现。这一发现通过最小描述长度原则，统一了学习动态与存储容量，构建了一个自组织临界性的几何理论。
原文摘要 · Abstract (English)
High-capacity kernel Hopfield networks exhibit a \textit{Ridge of Optimization} characterized by extreme stability. While previously linked to \textit{Spectral Concentration}, its origin remains elusive. Here, we analyze the network dynamics on a statistical manifold, revealing that the Ridge corresponds to the Edge of Stability, a critical boundary where the Fisher Information Matrix becomes singular. We demonstrate that the apparent Euclidean force antagonism is a manifestation of \textit{Dual Equilibrium} in the Riemannian space. This unifies learning dynamics and capacity via the Minimum Description Length principle, offering a geometric theory of self-organized criticality.
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