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

arXiv: https://arxiv.org/abs/2605.00366 · HTML: https://mlsanity.com/paper/2605.00366/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 1 May 2026.

## Builds on (most cited first, 9 of 9)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Kernel Logistic Regression Learning for High-Capacity Hopfield Networks](https://mlsanity.com/paper/2504.07633/) (2025)
- [Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks](https://mlsanity.com/paper/2511.13053/) (2025)
- [Quality of internal representation shapes learning performance in feedback neural networks](https://mlsanity.com/paper/2011.06066/) (2020)
- [Quantitative Attractor Analysis of High-Capacity Kernel Hopfield Networks](https://mlsanity.com/paper/2505.01218/) (2025)
- [Quantization robustness from dense representations of sparse functions in high-capacity kernel associative memory](https://mlsanity.com/paper/2604.20333/) (2026)

## Abstract (arXiv)

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 ≈ 16, while maintaining stable retrieval for structured data at effective loads near P/N ≈ 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.