Spectral Concentration at the Edge of Stability: Information Geometry of Kernel Associative Memory

2025-11-28

AI Summary

The origin of the “optimization ridge”—an extreme stability phenomenon in high-capacity kernel Hopfield networks—remains poorly understood. Method: Leveraging information geometry and dynamical modeling on statistical manifolds, we identify its root at the critical stability boundary—specifically, the “edge of stability” where the Fisher information matrix becomes singular. We introduce a dual-balance mechanism, proving that force competition in Euclidean space corresponds to intrinsic equilibrium on the Riemannian manifold. Integrating kernel methods, nonlinear dynamical systems analysis, and spectral concentration theory, we precisely characterize this stability boundary. Contribution/Results: We unify learning dynamics and memory capacity within a novel information-geometric framework for kernel associative memory, uniquely merging the Minimum Description Length principle with self-organized criticality. Our theory reveals that optimal information compression and memory performance are achieved precisely at the edge of stability, thereby resolving the mechanistic basis of the optimization ridge.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

High-capacity kernel Hopfield networks exhibit a "Ridge of Optimization" characterized by extreme stability. While previously linked to "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 extit{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.

Problem

• Unifying spectral concentration origins in high-capacity kernel Hopfield networks
• Analyzing network dynamics at critical Fisher Information Matrix singularity
• Explaining Euclidean force antagonism through Riemannian dual equilibrium

Innovation

• Analyzes network dynamics on statistical manifold
• Reveals Ridge corresponds to Edge of Stability
• Unifies learning via Minimum Description Length principle