Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Differential and Riemannian Calculus on Wasserstein Spaces

André Magalhães de Sá Gomes, Christian S. Rodrigues, Luiz A. B. San Martin

Abstract

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology via convergent sequences of the subjacent space. Applying it we also provide a new proof that Wasserstein spaces of closed manifolds are geodesically convex. Our framework is particularly handy to address the Wasserstein spaces of compact Lie groups, where we refine our formalism and present an explicit example.

On Differential and Riemannian Calculus on Wasserstein Spaces

Abstract

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology via convergent sequences of the subjacent space. Applying it we also provide a new proof that Wasserstein spaces of closed manifolds are geodesically convex. Our framework is particularly handy to address the Wasserstein spaces of compact Lie groups, where we refine our formalism and present an explicit example.
Paper Structure (13 sections, 17 theorems, 93 equations)

This paper contains 13 sections, 17 theorems, 93 equations.

Table of Contents

  1. The Wasserstein Spaces
  2. Geodesic Space Structure of $P_2(M)$ - Displacement Interpolations
  3. On General Displacement Interpolations
  4. Differential Calculus on Wasserstein spaces
  5. Tangent Superspaces and Derivatives of Curves
  6. Curves with constant velocity field.
  7. Tangent Spaces and Directional Derivatives
  8. Riemannian geometry
  9. Riemannian Metric
  10. Lie Bracket and Levi-Civita Connection
  11. Parallelizable Wasserstein Spaces
  12. Lie Bracket, Levi-Civita Connection and Geodesics
  13. Curvature

Key Result

Theorem 1.1

(Monge-Kantorovich) Let $M$ be Riemannian manifold with geodesic distance $d$ and take a pair $\mu,\nu\in P(M)$. Then, for $p\geq 1$, there is a coupling $\pi\in\Pi(\mu,\nu)$ which minimizes the optimal cost functional: Such a $\pi$ is called an optimal coupling between $\mu$ and $\nu$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • proof
  • Theorem 2.1
  • Proposition 2.2
  • ...and 16 more