Information Geometric Self-Organization at the Edge of Stability in High-Capacity Kernel Associative Memories

תקציר מקורי באנגליתarXiv:2609.16827v4 Announce Type: replace Abstract: High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit exceptional storage capabilities and robustness. Previous empirical studies identified a hyperparameter regime, the "Ridge of Optimization," where attractor stability is maximized. However, the geometric nature of this regime and the optimization dynamics required to reach it have remained unclear. In this paper, we investigate the static geometry of the parameter space and the learning trajectory of Gradient Descent (GD) in KLR-trained Hopfield networks. Using the eigenvalue spectrum of the Hessian, we reveal that the Ridge corresponds to a phase boundary located adjacent to a rank-1 spectral collapse, acting as a geometric singularity where the principa