# Inferring effective couplings with Restricted Boltzmann Machines

arXiv: https://arxiv.org/abs/2309.02292 · HTML: https://mlsanity.com/paper/2309.02292/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 60 references that are in the map.
- First posted on arXiv 5 Sep 2023.

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

- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Inverse statistical problems: from the inverse Ising problem to data science](https://mlsanity.com/paper/1702.01522/) (2017)
- [Restricted Boltzmann Machine, recent advances and mean-field theory](https://mlsanity.com/paper/2011.11307/) (2020)
- [Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2105.13889/) (2021)
- [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)
- [Emergence of Compositional Representations in Restricted Boltzmann Machines](https://mlsanity.com/paper/1611.06759/) (2016)
- [Explaining the effects of non-convergent sampling in the training of Energy-Based Models](https://mlsanity.com/paper/2301.09428/) (2023)
- [Learning a Restricted Boltzmann Machine using biased Monte Carlo sampling](https://mlsanity.com/paper/2206.01310/) (2022)
- [Searching for collective behavior in a network of real neurons](https://mlsanity.com/paper/1306.3061/) (2013)
- [Restricted Boltzmann Machines as Models of Interacting Variables](https://mlsanity.com/paper/2103.15917/) (2021)
- [Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics](https://mlsanity.com/paper/2307.06797/) (2023)
- [A Deterministic and Generalized Framework for Unsupervised Learning with Restricted Boltzmann Machines](https://mlsanity.com/paper/1702.03260/) (2017)

## Built on by (most cited first, 8 of 8)

- [Fast training and sampling of Restricted Boltzmann Machines](https://mlsanity.com/paper/2405.15376/) (2024)
- [A theoretical framework for overfitting in energy-based modeling](https://mlsanity.com/paper/2501.19158/) (2025)
- [The effect of priors on Learning with Restricted Boltzmann Machines](https://mlsanity.com/paper/2412.02623/) (2024)
- [On the role of non-linear latent features in bipartite generative neural networks](https://mlsanity.com/paper/2506.10552/) (2025)
- [Diluting Restricted Boltzmann Machines](https://mlsanity.com/paper/2511.00648/) (2025)
- [Distributional simplicity bias and effective convexity in Energy Based Models](https://mlsanity.com/paper/2605.07844/) (2026)
- [Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering](https://mlsanity.com/paper/2607.27077/) (2026)
- [Nonlocal Transition Kernel for Efficient Learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2608.17450/) (2026)

## Abstract (arXiv)

Generative models offer a direct way of modeling complex data. Energy-based models attempt to encode the statistical correlations observed in the data at the level of the Boltzmann weight associated with an energy function in the form of a neural network. We address here the challenge of understanding the physical interpretation of such models. In this study, we propose a simple solution by implementing a direct mapping between the Restricted Boltzmann Machine and an effective Ising spin Hamiltonian. This mapping includes interactions of all possible orders, going beyond the conventional pairwise interactions typically considered in the inverse Ising (or Boltzmann Machine) approach, and allowing the description of complex datasets. Earlier works attempted to achieve this goal, but the proposed mappings were inaccurate for inference applications, did not properly treat the complexity of the problem, or did not provide precise prescriptions for practical application. To validate our method, we performed several controlled inverse numerical experiments in which we trained the RBMs using equilibrium samples of predefined models with local external fields, 2-body and 3-body interactions in different sparse topologies. The results demonstrate the effectiveness of our proposed approach in learning the correct interaction network and pave the way for its application in modeling interesting binary variable datasets. We also evaluate the quality of the inferred model based on different training methods.