{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08612",
  "author_total": 3,
  "authors": [
    {
      "name": "Elena Agliari"
    },
    {
      "name": "Andrea Lepre"
    },
    {
      "name": "Edoardo Roscani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9hpecdd8bjejy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08612",
  "fulltext_url": null,
  "id": "9hpecdd8bjejy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08612",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7221064579",
  "pdf_url": "https://arxiv.org/pdf/2610.08612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9hpecdd8bjejy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.08612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08612"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Classifications in modular restricted Boltzmann machines",
  "title_text": "Classifications in modular restricted Boltzmann machines",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/9hpecdd8bjejy",
  "venue": "arXiv",
  "versions": []
}