Geometric analysis of attractor boundaries and storage capacity limits in kernel Hopfield networks

In a new preprint on arXiv, Akira Tamamori presents a geometric analysis of attractor boundaries and storage capacity limits in kernel Hopfield networks trained via Kernel Logistic Regression (KLR). The study combines empirical evaluations on random sequences and real-world image embeddings from CIFAR-10 with phenomenological morphing experiments and Signal-to-Noise Ratio (SNR) analysis. Results show the networks achieve a storage capacity of up to P/N ≈ 16 for random sequences and maintain stable retrieval at effective loads near P/N ≈ 20 for structured data. Morphing analysis reveals that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries characterized by steep effective potential barriers and critical slowing down.

By contrasting SNR analysis with a geometric reference point inspired by Cover's theorem, the author demonstrates that the ultimate storage limit is constrained primarily by loss of dynamical stability against crosstalk noise, not by a lack of geometric separability in feature space. These findings suggest KLR networks function as highly localized, exemplar-based memories operating optimally just before dynamical collapse. The work provides new theoretical insights for designing robust, large-scale retrieval systems in neural computing and associative memory applications.