# A remark on a paper of Krotov and Hopfield [arXiv:2008.06996]

arXiv: https://arxiv.org/abs/2105.15034 · HTML: https://mlsanity.com/paper/2105.15034/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 5 references that are in the map.
- First posted on arXiv 31 May 2021.

## Builds on (most cited first, 5 of 5)

- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [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)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Do You Even Need Attention? A Stack of Feed-Forward Layers Does Surprisingly Well on ImageNet](https://mlsanity.com/paper/2105.02723/) (2021)

## Built on by (most cited first, 11 of 11)

- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models](https://mlsanity.com/paper/2202.04557/) (2022)
- [Hierarchical Associative Memory](https://mlsanity.com/paper/2107.06446/) (2021)
- [Attention in a family of Boltzmann machines emerging from modern Hopfield networks](https://mlsanity.com/paper/2212.04692/) (2022)
- [Hierarchical Associative Memory, Parallelized MLP-Mixer, and Symmetry Breaking](https://mlsanity.com/paper/2406.12220/) (2024)
- [iMixer: hierarchical Hopfield network implies an invertible, implicit and iterative MLP-Mixer](https://mlsanity.com/paper/2304.13061/) (2023)
- [Associative memory and dead neurons](https://mlsanity.com/paper/2410.13866/) (2024)
- [Neural Learning Rules from Associative Networks Theory](https://mlsanity.com/paper/2503.19922/) (2025)
- [Classifying States of the Hopfield Network with Improved Accuracy, Generalization, and Interpretability](https://mlsanity.com/paper/2503.03018/) (2025)
- [Binding threshold units with artificial oscillatory neurons](https://mlsanity.com/paper/2505.03648/) (2025)
- [Controlled Dynamics Attractor Transformer](https://mlsanity.com/paper/2606.15207/) (2026)

## Abstract (arXiv)

In their recent paper titled "Large Associative Memory Problem in Neurobiology and Machine Learning" [arXiv:2008.06996] the authors gave a biologically plausible microscopic theory from which one can recover many dense associative memory models discussed in the literature. We show that the layers of the recent "MLP-mixer" [arXiv:2105.01601] as well as the essentially equivalent model in [arXiv:2105.02723] are amongst them.