Dynamical Optimal Transport on Discrete Surfaces

Hugo Lavenant (1) , Sebastian Claici (2) , Edward Chien (2) , Justin Solomon (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 MIT - Massachusetts Institute of Technology

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.

Mots clés: Shape analysis; Interpolation; Convex optimization; Partial differential equations; Optimal Transport; Wasserstein Distance; Discrete Differential Geometry

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [cs.NA]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Hugo Lavenant, Sebastian Claici, Edward Chien, Justin Solomon. Dynamical Optimal Transport on Discrete Surfaces. SIGGRAPH Asia 2018, Dec 2018, Tokyo, Japan. ⟨10.1145/3272127.3275064⟩. ⟨hal-01875836⟩