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

TL;DR

This paper explores the geometric and dynamical factors affecting storage capacity and stability in kernel-based Hopfield networks, revealing that stability loss, not geometric separability, limits memory capacity.

Contribution

It provides a detailed analysis of attractor boundaries and stability mechanisms in KLR-trained Hopfield networks, combining empirical, morphing, and SNR analyses.

Findings

Storage capacity for random sequences reaches P/N ≈ 16

Stable retrieval for structured data near P/N ≈ 20

Attractor boundaries exhibit sharp phase transitions and steep potential barriers

Abstract

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to $P/N \approx 16$, while maintaining stable retrieval for structured data at effective loads near $P/N \approx 20$. Morphing analysis indicates 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. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.