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JKO schemes with general transport costs

Cale Rankin, Ting-Kam Leonard Wong

Abstract

We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.

JKO schemes with general transport costs

Abstract

We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.
Paper Structure (17 sections, 13 theorems, 107 equations, 1 table)

This paper contains 17 sections, 13 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on optimal transport and the JKO scheme
  4. Preliminary analysis
  5. Lower semicontinuity and convexity of $J_k$
  6. On $c$-convex sets and $c$-segments
  7. Differentiation formulas
  8. Pre-compact submanifolds
  9. Existence of minimizer
  10. Minimality condition
  11. Derivation of the Fokker--Planck equation
  12. Complete manifold
  13. Existence of a minimizer
  14. Derivation of the Fokker--Planck equation
  15. The Bregman case
  16. ...and 2 more sections

Key Result

Theorem 1

Assume $c$ is a cost function on $N$ satisfying eq:cg-def as well as conditions A1, and A2. Assume further $N$ is $c$-convex with respect to itself and that there are constants $\lambda,\Lambda > 0$ such that $\lambda d^2(x,y) \leq c(x,y) \leq \Lambda d^2(x,y)$ for $x, y \in M$. Assume either: Then there exists a measurable function $\rho: [0,\infty) \times M \rightarrow\mathbf{R}_+$ such that fo

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Initial velocity of $c$-segment
  • proof
  • Lemma 4
  • proof
  • ...and 16 more