{
  "id": "2303.06375",
  "version": "v1",
  "published": "2023-03-11T10:54:13.000Z",
  "updated": "2023-03-11T10:54:13.000Z",
  "title": "About the de Almeida-Thouless line in neural networks",
  "authors": [
    "Linda Albanese",
    "Adriano Barra",
    "Alessia Annibale"
  ],
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "In this work we present a rigorous and straightforward method to detect the onset of the instability of replica-symmetric theories in information processing systems, which does not require a full replica analysis as in the method originally proposed by Almeida-Thouless for spin glasses. The method is based on an expansion of the free-energy obtained within one-step of the replica symmetry breaking (RSB) scheme around the replica-symmetric (RS) value. As such, it requires solely continuity and differentiability of the free-energy and it is robust to be applied broadly to systems with quenched disorder. We apply the method to the Hopfield model and to neural networks with multi-node Hebbian interactions, as case studies. In the appendix we test the method on the Sherrington-Kirkpatrick, the Ising P-spin and the spherical P-spin models, recovering the AT lines known in the literature for these models, as a special limit, which corresponds to assuming that the transition from the RS to the RSB phase can be obtained by varying continuously the order parameters. Therefore, our method provides a generalization of the AT approach, which can be applied to systems with a discontinuous transition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-11T10:54:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "neural networks",
      "almeida-thouless line",
      "full replica analysis",
      "multi-node hebbian interactions",
      "spin glasses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}