{
  "id": "1902.05451",
  "version": "v1",
  "published": "2019-02-14T15:41:47.000Z",
  "updated": "2019-02-14T15:41:47.000Z",
  "title": "The Wasserstein Distances Between Pushed-Forward Measures with Applications to Uncertainty Quantification",
  "authors": [
    "Amir Sagiv"
  ],
  "categories": [
    "math.CA",
    "math.NA",
    "math.PR"
  ],
  "abstract": "In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\\varrho$. The system's response $f$ pushes forward $\\varrho$ to a new measure $f\\circ \\varrho$ which we would like to study. However, we might not have access to $f$ but only to its approximation $g$. We thus arrive at a fundamental question -- if $f$ and $g$ are close in $L^q$, does $g\\circ \\varrho$ approximate $f\\circ \\varrho$ well, and in what sense? Previously, we demonstrated that the answer to this question might be negative in terms of the $L^p$ distance between probability density functions (PDF). Here we show that the Wasserstein metric is the proper framework for this question. For any $p\\geq 1$, we bound the Wasserstein distance $W_p (f\\circ \\varrho , g\\circ \\varrho) $ from above by $\\|f-g\\|_{q}$. Furthermore, we provide lower bounds for the cases of $p=1,2$. Finally, we apply our theory to the analysis of common numerical methods in the field of computational uncertainty quantification.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-14T15:41:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A10",
      "60A10",
      "65D99"
    ],
    "keywords": [
      "wasserstein distance",
      "pushed-forward measures",
      "applications",
      "probability density functions",
      "computational uncertainty quantification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}