{
  "id": "2006.06994",
  "version": "v1",
  "published": "2020-06-12T08:15:20.000Z",
  "updated": "2020-06-12T08:15:20.000Z",
  "title": "Sparse approximation of triangular transports on bounded domains",
  "authors": [
    "Jakob Zech",
    "Youssef Marzouk"
  ],
  "categories": [
    "math.NA",
    "cs.NA",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $\\rho$ and $\\pi$ be two probability measures on $[-1,1]^d$ with positive and analytic Lebesgue densities. We investigate the approximation of the unique triangular monotone (Knothe-Rosenblatt) transport $T:[-1,1]^d\\to [-1,1]^d$, such that the pushforward $T_\\sharp\\rho$ equals $\\pi$. It is shown that for $d\\in\\mathbb{N}$ there exist approximations $\\tilde T$ of $T$ based on either sparse polynomial expansions or ReLU networks, such that the distance between $\\tilde T_\\sharp\\rho$ and $\\pi$ decreases exponentially. More precisely, we show error bounds of the type $\\exp(-\\beta N^{1/d})$ (or $\\exp(-\\beta N^{1/(d+1)})$ for neural networks), where $N$ refers to the dimension of the ansatz space (or the size of the network) containing $\\tilde T$; the notion of distance comprises, among others, the Hellinger distance and the Kullback--Leibler divergence. The construction guarantees $\\tilde T$ to be a monotone triangular bijective transport on the hypercube $[-1,1]^d$. Analogous results hold for the inverse transport $S=T^{-1}$. The proofs are constructive, and we give an explicit a priori description of the ansatz space, which can be used for numerical implementations. Additionally we discuss the high-dimensional case: for $d=\\infty$ a dimension-independent algebraic convergence rate is proved for a class of probability measures occurring widely in Bayesian inference for uncertainty quantification, thus verifying that the curse of dimensionality can be overcome in this setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-12T08:15:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32D05",
      "41A10",
      "41A25",
      "41A46",
      "62D99",
      "65D15"
    ],
    "keywords": [
      "sparse approximation",
      "triangular transports",
      "bounded domains",
      "ansatz space",
      "probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}