# Phases in a class of associative memories via hidden neurons

arXiv: https://arxiv.org/abs/2609.10976 · HTML: https://mlsanity.com/paper/2609.10976/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 59 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Hierarchical Associative Memory](https://mlsanity.com/paper/2107.06446/) (2021)
- [Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors](https://mlsanity.com/paper/1702.05882/) (2017)
- [Provably Optimal Memory Capacity for Modern Hopfield Models: Transformer-Compatible Dense Associative Memories as Spherical Codes](https://mlsanity.com/paper/2410.23126/) (2024)
- [Attention in a family of Boltzmann machines emerging from modern Hopfield networks](https://mlsanity.com/paper/2212.04692/) (2022)
- [A Biologically Plausible Dense Associative Memory with Exponential Capacity](https://mlsanity.com/paper/2601.00984/) (2026)
- [Hierarchical Associative Memory, Parallelized MLP-Mixer, and Symmetry Breaking](https://mlsanity.com/paper/2406.12220/) (2024)
- [Replica Symmetry Breaking in Bipartite Spin Glasses and Neural Networks](https://mlsanity.com/paper/1803.06442/) (2018)
- [Exploring the Temperature-Dependent Phase Transition in Modern Hopfield Networks](https://mlsanity.com/paper/2311.18434/) (2023)
- [Geometric Entropy and Retrieval Phase Transitions in Continuous Thermal Dense Associative Memory](https://mlsanity.com/paper/2604.07401/) (2026)

## Abstract (arXiv)

Associative memory in the Hopfield network is attractor dynamics in a disordered many-body system, and higher-order and exponential extensions turn its retrieval update into softmax attention. The polynomial and exponential regimes have been analyzed by different methods, with no common architecture in which to ask what fixes the storage scale. In this paper we study the bipartite architecture of Krotov and Hopfield, which we call the class H, whose model is fixed by a Lagrangian for each layer, taking the hidden neurons as the order parameter of retrieval. At polynomial load the replica method yields the replica-symmetric phase diagrams and closed-form capacities, and the crosstalk moment is common to Ising and spherical visible neurons, so their differences come from the visible entropy. With a softmax hidden layer the load is exponential, and a copy representation maps the thermodynamics onto random-energy-model counting, with paramagnetic, condensed, and frozen phases. Heating destabilizes retrieval by quantized reassignments of attention, and typical Gaussian patterns remain metastable at every load. The regimes differ in their crosstalk statistics, central-limit at polynomial load and large-deviation at exponential load, and the class H splits retrieval into two roles, the visible Lagrangian fixing stability and the hidden one the storage scale, two axes that may also guide the design of new Lagrangians.