{
  "id": "1507.02803",
  "version": "v1",
  "published": "2015-07-10T08:20:49.000Z",
  "updated": "2015-07-10T08:20:49.000Z",
  "title": "Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance",
  "authors": [
    "Katalin Marton"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For a fixed probability measure $q^n$ on $\\mathcal X^n$, ($\\mathcal X$ is a finite set), and any probability measure $p^n=\\mathcal L(Y^n)$ on $\\mathcal X^n$, we have \\begin{equation}\\label{*} D(p^n||q^n)\\leq Const. \\sum_{i=1}^n \\Bbb E_{p^n} D(p_i(\\cdot|Y_1,\\dots, Y_{i-1},Y_{i+1},\\dots, Y_n) || q_i(\\cdot|Y_1,\\dots, Y_{i-1},Y_{i+1},\\dots, Y_n)), \\end{equation} where $p_i(\\cdot|y_1,\\dots, y_{i-1},y_{i+1},\\dots, y_n)$ and $q_i(\\cdot|x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)$ denote the local specifications for $p^n$ resp. $q^n$. The constant shall depend on the properties of the local specifications of $q^n$. Inequality (*) is meaningful in product spaces, both in the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for $q^n$, provided uniform logarithmic Sobolev inequalities are available for $q_i(\\cdot|x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)$, for all fixed $i$ and all fixed $(x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)$. Inequality (*) directly implies that the Gibbs sampler associated with $q^n$ is a contraction for relative entropy. We derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking, a contractivity property of relative entropy with respect to some Markov semigroup. It is much easier to prove contractivity for a distance between measures than for relative entropy, since distances satisfy the triangle inequality, and for them well known linear tools, like estimates through matrix norms can be applied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-10T08:20:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C22",
      "60J05",
      "35Q84",
      "60J25",
      "82B21"
    ],
    "keywords": [
      "logarithmic sobolev inequality",
      "discrete product spaces",
      "transportation cost distance",
      "relative entropy",
      "uniform logarithmic sobolev inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150702803M"
    }
  }
}