# A Dynamical Mean-Field Theory for Learning in Restricted Boltzmann Machines

arXiv: https://arxiv.org/abs/2005.01560 · HTML: https://mlsanity.com/paper/2005.01560/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2020.

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

- [Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models](https://mlsanity.com/paper/2212.01572/) (2022)
- [Fundamental operating regimes, hyper-parameter fine-tuning and glassiness: towards an interpretable replica-theory for trained restricted Boltzmann machines](https://mlsanity.com/paper/2406.09924/) (2024)
- [Analysis of Random Sequential Message Passing Algorithms for Approximate Inference](https://mlsanity.com/paper/2202.08198/) (2022)

## Abstract (arXiv)

We define a message-passing algorithm for computing magnetizations in Restricted Boltzmann machines, which are Ising models on bipartite graphs introduced as neural network models for probability distributions over spin configurations. To model nontrivial statistical dependencies between the spins' couplings, we assume that the rectangular coupling matrix is drawn from an arbitrary bi-rotation invariant random matrix ensemble. Using the dynamical functional method of statistical mechanics we exactly analyze the dynamics of the algorithm in the large system limit. We prove the global convergence of the algorithm under a stability criterion and compute asymptotic convergence rates showing excellent agreement with numerical simulations.