# The effect of priors on Learning with Restricted Boltzmann Machines

arXiv: https://arxiv.org/abs/2412.02623 · HTML: https://mlsanity.com/paper/2412.02623/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 3 Dec 2024.

## Builds on (most cited first, 21 of 21)

- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [Restricted Boltzmann Machine, recent advances and mean-field theory](https://mlsanity.com/paper/2011.11307/) (2020)
- [Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors](https://mlsanity.com/paper/1702.05882/) (2017)
- [Thermodynamics of Restricted Boltzmann Machines and related learning dynamics](https://mlsanity.com/paper/1803.01960/) (2018)
- [Phase transitions in Restricted Boltzmann Machines with generic priors](https://mlsanity.com/paper/1612.03132/) (2016)
- [Minimal model of permutation symmetry in unsupervised learning](https://mlsanity.com/paper/1904.13052/) (2019)
- [Cascade of phase transitions in the training of Energy-based models](https://mlsanity.com/paper/2405.14689/) (2024)
- [Statistical mechanics of unsupervised feature learning in a restricted Boltzmann machine with binary synapses](https://mlsanity.com/paper/1612.01717/) (2016)
- [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)
- [Efficiency of quantum versus classical annealing in non-convex learning problems](https://mlsanity.com/paper/1706.08470/) (2017)
- [Inferring effective couplings with Restricted Boltzmann Machines](https://mlsanity.com/paper/2309.02292/) (2023)
- [Hopfield model with planted patterns: a teacher-student self-supervised learning model](https://mlsanity.com/paper/2304.13710/) (2023)
- [Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics](https://mlsanity.com/paper/2307.06797/) (2023)
- [Unsupervised feature learning from finite data by message passing: discontinuous versus continuous phase transition](https://mlsanity.com/paper/1608.03714/) (2016)
- [Approximate Survey Propagation for Statistical Inference](https://mlsanity.com/paper/1807.01296/) (2018)
- [From Dyson to Hopfield: Processing on hierarchical networks](https://mlsanity.com/paper/1407.5019/) (2014)
- [Thermodynamics of bidirectional associative memories](https://mlsanity.com/paper/2211.09694/) (2022)
- [Dense Hopfield Networks in the Teacher-Student Setting](https://mlsanity.com/paper/2401.04191/) (2024)
- [Bayes-optimal limits in structured PCA, and how to reach them](https://mlsanity.com/paper/2210.01237/) (2022)
- [Modeling Structured Data Learning with Restricted Boltzmann Machines in the Teacher-Student Setting](https://mlsanity.com/paper/2410.16150/) (2024)

## Built on by (most cited first, 3 of 3)

- [Learning with Restricted Boltzmann Machines: Asymptotics of AMP and GD in High Dimensions](https://mlsanity.com/paper/2505.18046/) (2025)
- [On the role of non-linear latent features in bipartite generative neural networks](https://mlsanity.com/paper/2506.10552/) (2025)
- [Activation Functions, Statistics and Learning of Higher-Order Interactions in Restricted Boltzmann Machines](https://mlsanity.com/paper/2605.19178/) (2026)

## Abstract (arXiv)

Restricted Boltzmann Machines (RBMs) are generative models designed to learn from data with a rich underlying structure. In this work, we explore a teacher-student setting where a student RBM learns from examples generated by a teacher RBM, with a focus on the effect of the unit priors on learning efficiency. We consider a parametric class of priors that interpolate between continuous (Gaussian) and binary variables. This approach models various possible choices of visible units, hidden units, and weights for both the teacher and student RBMs. By analyzing the phase diagram of the posterior distribution in both the Bayes optimal and mismatched regimes, we demonstrate the existence of a triple point that defines the critical dataset size necessary for learning through generalization. The critical size is strongly influenced by the properties of the teacher, and thus the data, but is unaffected by the properties of the student RBM. Nevertheless, a prudent choice of student priors can facilitate training by expanding the so-called signal retrieval region, where the machine generalizes effectively.