Dynamical Optimal Transport on Discrete Surfaces

TL;DR

This paper introduces a novel dynamical optimal transport method for discrete surfaces that preserves structure and enables smooth, low-diffusion interpolation between probability distributions, extending to various applications.

Contribution

It adapts a continuous dynamical optimal transport framework to discrete surfaces, creating a structure-preserving Riemannian metric on probability distributions.

Findings

Provides smoother interpolation with less diffusion than existing methods.

Extends optimal transport to Dirichlet problems and gradient flow integration on discrete surfaces.

Demonstrates practical applicability in distribution interpolation and related tasks.

Abstract

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.