Classifications in modular restricted Boltzmann machines
Abstract
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.
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BibTeX
@misc{agliari2026classifications,
title = {{Classifications in modular restricted Boltzmann machines}},
author = {Elena Agliari and Andrea Lepre and Edoardo Roscani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08612},
eprint = {2610.08612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08612},
}
RIS
TY - UNPB
TI - Classifications in modular restricted Boltzmann machines
AU - Elena Agliari
AU - Andrea Lepre
AU - Edoardo Roscani
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08612
UR - https://arxiv.org/abs/2610.08612
AB - 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.
ER -