# Restricted Boltzmann Machines as Models of Interacting Variables

arXiv: https://arxiv.org/abs/2103.15917 · HTML: https://mlsanity.com/paper/2103.15917/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 29 Mar 2021.

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

- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Shaping the learning landscape in neural networks around wide flat minima](https://mlsanity.com/paper/1905.07833/) (2019)
- [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)
- [Effect of Activation Functions on the Training of Overparametrized Neural Nets](https://mlsanity.com/paper/1908.05660/) (2019)
- [Efficiency of local learning rules in threshold-linear associative networks](https://mlsanity.com/paper/2007.12584/) (2020)
- [Learning Compositional Representations of Interacting Systems with Restricted Boltzmann Machines: Comparative Study of Lattice Proteins](https://mlsanity.com/paper/1902.06495/) (2019)
- [Learning with hidden variables](https://mlsanity.com/paper/1506.00354/) (2015)
- [Stochastic Neural Networks with Monotonic Activation Functions](https://mlsanity.com/paper/1601.00034/) (2016)
- [Modeling Retinal Ganglion Cell Population Activity with Restricted Boltzmann Machines](https://mlsanity.com/paper/1701.02898/) (2017)
- [Blindfold learning of an accurate neural metric](https://mlsanity.com/paper/1710.05067/) (2017)

## Built on by (most cited first, 9 of 9)

- [Fast training and sampling of Restricted Boltzmann Machines](https://mlsanity.com/paper/2405.15376/) (2024)
- [Inferring effective couplings with Restricted Boltzmann Machines](https://mlsanity.com/paper/2309.02292/) (2023)
- [Inferring Higher-Order Couplings with Neural Networks](https://mlsanity.com/paper/2501.06108/) (2025)
- [Quantifying Relevance in Learning and Inference](https://mlsanity.com/paper/2202.00339/) (2022)
- [On the role of non-linear latent features in bipartite generative neural networks](https://mlsanity.com/paper/2506.10552/) (2025)
- [Pattern reconstruction with restricted Boltzmann machines](https://mlsanity.com/paper/2205.07087/) (2022)
- [Diluting Restricted Boltzmann Machines](https://mlsanity.com/paper/2511.00648/) (2025)
- [Activation Functions, Statistics and Learning of Higher-Order Interactions in Restricted Boltzmann Machines](https://mlsanity.com/paper/2605.19178/) (2026)
- [Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering](https://mlsanity.com/paper/2607.27077/) (2026)

## Abstract (arXiv)

We study the type of distributions that Restricted Boltzmann Machines (RBMs) with different activation functions can express by investigating the effect of the activation function of the hidden nodes on the marginal distribution they impose on observed binary nodes. We report an exact expression for these marginals in the form of a model of interacting binary variables with the explicit form of the interactions depending on the hidden node activation function. We study the properties of these interactions in detail and evaluate how the accuracy with which the RBM approximates distributions over binary variables depends on the hidden node activation function and on the number of hidden nodes. When the inferred RBM parameters are weak, an intuitive pattern is found for the expression of the interaction terms which reduces substantially the differences across activation functions. We show that the weak parameter approximation is a good approximation for different RBMs trained on the MNIST dataset. Interestingly, in these cases, the mapping reveals that the inferred models are essentially low order interaction models.