# Unsupervised and Supervised learning by Dense Associative Memory under replica symmetry breaking

arXiv: https://arxiv.org/abs/2312.09638 · HTML: https://mlsanity.com/paper/2312.09638/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 15 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [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)
- [An exact mapping between the Variational Renormalization Group and Deep Learning](https://mlsanity.com/paper/1410.3831/) (2014)
- [Dense Associative Memory is Robust to Adversarial Inputs](https://mlsanity.com/paper/1701.00939/) (2017)
- [Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors](https://mlsanity.com/paper/1702.05882/) (2017)
- [Emergence of Compositional Representations in Restricted Boltzmann Machines](https://mlsanity.com/paper/1611.06759/) (2016)
- [Phase transitions in Restricted Boltzmann Machines with generic priors](https://mlsanity.com/paper/1612.03132/) (2016)
- [The emergence of a concept in shallow neural networks](https://mlsanity.com/paper/2109.00454/) (2021)
- [Supervised Hebbian Learning](https://mlsanity.com/paper/2203.01304/) (2022)
- [Neural networks with redundant representation: detecting the undetectable](https://mlsanity.com/paper/1911.12689/) (2019)
- [Dense Hebbian neural networks: a replica symmetric picture of unsupervised learning](https://mlsanity.com/paper/2211.14067/) (2022)
- [Dense Hebbian neural networks: a replica symmetric picture of supervised learning](https://mlsanity.com/paper/2212.00606/) (2022)
- [Parallel Learning by Multitasking Neural Networks](https://mlsanity.com/paper/2308.04106/) (2023)

## Abstract (arXiv)

Statistical mechanics of spin glasses is one of the main strands toward a comprehension of information processing by neural networks and learning machines. Tackling this approach, at the fairly standard replica symmetric level of description, recently Hebbian attractor networks with multi-node interactions (often called Dense Associative Memories) have been shown to outperform their classical pairwise counterparts in a number of tasks, from their robustness against adversarial attacks and their capability to work with prohibitively weak signals to their supra-linear storage capacities. Focusing on mathematical techniques more than computational aspects, in this paper we relax the replica symmetric assumption and we derive the one-step broken-replica-symmetry picture of supervised and unsupervised learning protocols for these Dense Associative Memories: a phase diagram in the space of the control parameters is achieved, independently, both via the Parisi's hierarchy within then replica trick as well as via the Guerra's telescope within the broken-replica interpolation. Further, an explicit analytical investigation is provided to deepen both the big-data and ground state limits of these networks as well as a proof that replica symmetry breaking does not alter the thresholds for learning and slightly increases the maximal storage capacity. Finally the De Almeida and Thouless line, depicting the onset of instability of a replica symmetric description, is also analytically derived highlighting how, crossed this boundary, the broken replica description should be preferred.