Spectral Concentration at the Edge of Stability: Information Geometry of Kernel Associative Memory
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Abstract (arXiv)
High-capacity kernel Hopfield networks exhibit a textitRidge of Optimization characterized by extreme stability. While previously linked to textitSpectral 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 textitDual 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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References · 5
- 2020Prevalence of Neural Collapse during the terminal phase of deep learning trainingcited by 719
- 2020Hopfield Networks is All You Needcited by 475
- 2021Gradient Descent on Neural Networks Typically Occurs at the Edge of Stabilitycited by 427
- 2018Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approachcited by 91
- 2025Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networkscited by 5
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