{
  "id": "1805.06982",
  "version": "v1",
  "published": "2018-05-17T22:26:08.000Z",
  "updated": "2018-05-17T22:26:08.000Z",
  "title": "A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise",
  "authors": [
    "Yan V Fyodorov"
  ],
  "comment": "33 pages, 5 figures",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "An encryption of a signal ${\\bf s}\\in\\mathbb{R^N}$ is a random mapping ${\\bf s}\\mapsto \\textbf{y}=(y_1,\\ldots,y_M)^T\\in \\mathbb{R}^M$ which can be corrupted by an additive noise. Given the Encryption Redundancy Parameter (ERP) $\\mu=M/N\\ge 1$, the signal strength parameter $R=\\sqrt{\\sum_i s_i^2/N}$, and the ('bare') noise-to-signal ratio (NSR) $\\gamma\\ge 0$, we consider the problem of reconstructing ${\\bf s}$ from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian-distributed random potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap $p_{\\infty}\\in [0,1]$ between the original signal and its recovered image (known as 'estimator') as $N\\to \\infty$, which is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full $p_{\\infty} (\\gamma)$ curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with $p_{\\infty}>0$ for any $\\mu>1$ and any $\\gamma<\\infty$, with $p_{\\infty}\\sim \\gamma^{-1/2}$ as $\\gamma\\to \\infty$. In contrast, for the case of purely quadratic nonlinearity, for any ERP $\\mu>1$ there exists a threshold NSR value $\\gamma_c(\\mu)$ such that $p_{\\infty}=0$ for $\\gamma>\\gamma_c(\\mu)$ making the reconstruction impossible. The behaviour close to the threshold is given by $p_{\\infty}\\sim (\\gamma_c-\\gamma)^{3/4}$ and is controlled by the replica symmetry breaking mechanism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-17T22:26:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spin glass model",
      "reconstructing nonlinearly encrypted signals",
      "parisi replica symmetry breaking scheme",
      "nonvanishing linear component permit reconstructions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}