# Free energy landscape of Dense Associative Memory

arXiv: https://arxiv.org/abs/2607.19195 · HTML: https://mlsanity.com/paper/2607.19195/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 16 references that are in the map.
- First posted on arXiv 21 Jul 2026.

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

- [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)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Hopfield Networks as Models of Emergent Function in Biology](https://mlsanity.com/paper/2506.13076/) (2025)

## Abstract (arXiv)

Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.