# Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks

arXiv: https://arxiv.org/abs/2511.13053 · HTML: https://mlsanity.com/paper/2511.13053/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 28 references that are in the map.
- First posted on arXiv 17 Nov 2025.

## Builds on (most cited first, 4 of 4)

- [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)
- [Quantitative Attractor Analysis of High-Capacity Kernel Hopfield Networks](https://mlsanity.com/paper/2505.01218/) (2025)

## Built on by (most cited first, 5 of 5)

- [Quantization robustness from dense representations of sparse functions in high-capacity kernel associative memory](https://mlsanity.com/paper/2604.20333/) (2026)
- [Spectral Concentration at the Edge of Stability: Information Geometry of Kernel Associative Memory](https://mlsanity.com/paper/2511.23083/) (2025)
- [Geometric and dynamical analysis of attractor boundaries and storage limits in kernel Hopfield networks](https://mlsanity.com/paper/2605.00366/) (2026)
- [Geometry of learning dynamics: Gradient descent versus natural gradient on the ridge of optimization](https://mlsanity.com/paper/2609.16805/) (2026)
- [Information Geometric Self-Organization at the Edge of Stability in High-Capacity Kernel Associative Memories](https://mlsanity.com/paper/2609.16827/) (2026)

## Abstract (arXiv)

Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanisms behind this enhancement remain poorly understood. We address this gap by combining a geometric characterization of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, Pinnacle Sharpness, we empirically uncover a rich phase diagram of attractor stability, identifying a Ridge of Optimization where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a Force Antagonism, in which a strong driving force is counterbalanced by a collective feedback force. We theoretically interpret this behavior as a consequence of a specific reorganization of the weight spectrum, which we term Spectral Concentration. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical regime: the leading eigenvalue is amplified to enhance global stability (Direct Force), while the trailing eigenvalues remain finite to sustain high memory capacity (Indirect Force). Together, these results suggest a spectral mechanism by which learning reconciles stability and capacity in high-dimensional associative memory models.