# Kernel Ridge Regression for Efficient Learning of High-Capacity Hopfield Networks

arXiv: https://arxiv.org/abs/2504.12561 · HTML: https://mlsanity.com/paper/2504.12561/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 17 Apr 2025.

## Builds on (most cited first, 3 of 3)

- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Kernel Memory Networks: A Unifying Framework for Memory Modeling](https://mlsanity.com/paper/2208.09416/) (2022)
- [Kernel Logistic Regression Learning for High-Capacity Hopfield Networks](https://mlsanity.com/paper/2504.07633/) (2025)

## Built on by (most cited first, 1 of 1)

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

## Abstract (arXiv)

Hopfield networks using Hebbian learning suffer from limited storage capacity. While supervised methods like Linear Logistic Regression (LLR) offer some improvement, kernel methods like Kernel Logistic Regression (KLR) significantly enhance storage capacity and noise robustness. However, KLR requires computationally expensive iterative learning. We propose Kernel Ridge Regression (KRR) as an efficient kernel-based alternative for learning high-capacity Hopfield networks. KRR utilizes the kernel trick and predicts bipolar states via regression, crucially offering a non-iterative, closed-form solution for learning dual variables. We evaluate KRR and compare its performance against Hebbian, LLR, and KLR. Our results demonstrate that KRR achieves state-of-the-art storage capacity (reaching a storage load of 1.5) and noise robustness, comparable to KLR. Crucially, KRR drastically reduces training time, being orders of magnitude faster than LLR and significantly faster than KLR, especially at higher storage loads. This establishes KRR as a potent and highly efficient method for building high-performance associative memories, providing comparable performance to KLR with substantial training speed advantages. This work provides the first empirical comparison between KRR and KLR in the context of Hopfield network learning.