# Classifications in modular restricted Boltzmann machines

arXiv: https://arxiv.org/abs/2610.08612 · HTML: https://mlsanity.com/paper/2610.08612/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 20 references that are in the map.
- First posted on arXiv 6 Oct 2026.

## Builds on (most cited first, 1 of 1)

- [On the mapping between Hopfield networks and Restricted Boltzmann Machines](https://mlsanity.com/paper/2101.11744/) (2021)

## Abstract (arXiv)

We consider a modular associative neural network made of L Hopfield models (HMs), coupled so that intra-module interactions are Hebbian and inter-module interactions are anti-Hebbian; this competitive coupling is known to endow the network with pattern-disentanglement capabilities. The integral representation of this system coincides with an assembly of L restricted Boltzmann machines (RBMs) whose hidden layers are coupled, thereby extending the HM-RBM duality to the modular setting. We then train this modular RBM, via one-step contrastive divergence, to perform a classification task in which a query encoded on the visible layers is mapped onto an L-tuple of labels read off the hidden layers. When the query is composed of L patterns that are mutually orthogonal on average, we prove that the RBM weights obtained as empirical means over the training dataset, as suggested by the HM-RBM equivalence, constitute a fixed point of the learning dynamics, and we derive an explicit, non-asymptotic bound on the residual drift. We then turn to a harder classification task in which each module is queried with a mixture of L patterns and we show numerically that the same setting for the RBM weights still provides an effective set-up, letting the network jointly classify and disentangle the mixture.