On Properties of the Generalized Wasserstein Distance

By Benedetto Piccoli, Francesco Rossi. Archive for Rational Mechanics and Analysis, 2013

117 citations recorded by the source corpus.

The Wasserstein distances Wp (p ≧\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\geqq}$$\end{document} 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W2 as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance W1(μ, ν) of two probability measures μ, ν and the supremum of the integrals in d(μ − ν) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances Wpa,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_p^{a,b}}$$\end{document}, defined in terms of both the classical Wasserstein distance Wp and the total variation (or L1) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here p plays the same role as for the classic Wasserstein distance, while a and b are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between W2a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_2^{a,b}}$$\end{document} and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality à la Kantorovich–Rubinstein establishing the equality between W11,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_1^{1,1}}$$\end{document} and the flat metric.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/28a3dc14e2c45f7f34e3389817d828de797ee437

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Lénaïc Chizat
  2. Bernhard Schmitzer
  3. Léonard Monsaingeon
  4. Dmitry Vorotnikov
  5. G. Peyr'e
  6. Stanislav Kondratyev
  7. François-Xavier Vialard
  8. Qing Nie

In Geometry and Topology

  1. Léonard Monsaingeon
  2. Dmitry Vorotnikov
  3. G. Peyr'e
  4. Bernhard Schmitzer
  5. Stanislav Kondratyev
  6. François-Xavier Vialard
  7. Lénaïc Chizat

Citations in the corpus