{
  "id": "0910.2281",
  "version": "v2",
  "published": "2009-10-13T12:50:43.000Z",
  "updated": "2009-11-21T03:55:30.000Z",
  "title": "Weight space structure and analysis using a finite replica number in the Ising perceptron",
  "authors": [
    "Tomoyuki Obuchi",
    "Yoshiyuki Kabashima"
  ],
  "comment": "21 pages, 11 figures, Added references, some comments, and corrections to minor errors",
  "journal": "J. Stat. Mech. (2009) P12014",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "The weight space of the Ising perceptron in which a set of random patterns is stored is examined using the generating function of the partition function $\\phi(n)=(1/N)\\log [Z^n]$ as the dimension of the weight vector $N$ tends to infinity, where $Z$ is the partition function and $[ ... ]$ represents the configurational average. We utilize $\\phi(n)$ for two purposes, depending on the value of the ratio $\\alpha=M/N$, where $M$ is the number of random patterns. For $\\alpha < \\alpha_{\\rm s}=0.833 ...$, we employ $\\phi(n)$, in conjunction with Parisi's one-step replica symmetry breaking scheme in the limit of $n \\to 0$, to evaluate the complexity that characterizes the number of disjoint clusters of weights that are compatible with a given set of random patterns, which indicates that, in typical cases, the weight space is equally dominated by a single large cluster of exponentially many weights and exponentially many small clusters of a single weight. For $\\alpha > \\alpha_{\\rm s}$, on the other hand, $\\phi(n)$ is used to assess the rate function of a small probability that a given set of random patterns is atypically separable by the Ising perceptrons. We show that the analyticity of the rate function changes at $\\alpha = \\alpha_{\\rm GD}=1.245 ... $, which implies that the dominant configuration of the atypically separable patterns exhibits a phase transition at this critical ratio. Extensive numerical experiments are conducted to support the theoretical predictions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-21T03:55:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weight space structure",
      "finite replica number",
      "ising perceptron",
      "random patterns",
      "partition function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1742-5468/2009/12/P12014",
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2009,
      "month": "Dec",
      "volume": 2009,
      "number": 12,
      "pages": 12014
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009JSMTE..12..014O"
    }
  }
}