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Strong equivalence between metrics of Wasserstein type

Erhan Bayraktar, Gaoyue Guo

Abstract

We appended an errata to the original submission. The purpose of this errata is to point out two errors in [2] and give a weakened version of those statements made.

Strong equivalence between metrics of Wasserstein type

Abstract

We appended an errata to the original submission. The purpose of this errata is to point out two errors in [2] and give a weakened version of those statements made.

Paper Structure

This paper contains 9 sections, 7 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries on the Wasserstein Metric
  4. Projected Wasserstein Metrics
  5. Main Results
  6. Proof of the Main Results
  7. Proof of Theorem \ref{['thm:equivalence']} (i)
  8. Proof of Theorem \ref{['thm:equivalence']} (ii)
  9. Proof of Theorem \ref{['thm:equivalence']} (iii)

Key Result

Proposition 2.1

(i) Fix $\mu, \nu\in{\mathcal{P}}_p(\mathbb{R}^d)$. The maps $\mathbb{S}^{d-1}\ni\mathsf{v}\mapsto \mu_{\mathsf{v}}\in {\mathcal{P}}_p(\mathbb{R})$ and $\mathbb{S}^{d-1}\ni\mathsf{v}\mapsto{\mathcal{W}}_p(\mu_{\mathsf{v}},\nu_{\mathsf{v}})\in\mathbb{R}_+$ are both Lipschitz, with respectively Lipsch

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Remark 2.1
  • proof : of Theorem \ref{['thm:equivalence']} (i)
  • Remark 3.1
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • ...and 10 more