{
  "id": "2207.07427",
  "version": "v1",
  "published": "2022-07-15T12:09:04.000Z",
  "updated": "2022-07-15T12:09:04.000Z",
  "title": "Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences",
  "authors": [
    "Alberto Gonzalez-Sanz",
    "Jean-Michel Loubes",
    "Jonathan Niles-Weed"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in $\\Cs$. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \\cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\\ {\\rm P}={\\rm Q}$ or $H_1:\\ {\\rm P}\\neq{\\rm Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-15T12:09:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "entropy regularized optimal transport",
      "weak limit",
      "entropic regularized optimal transport",
      "potentials",
      "gaussian process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}