Dynamical optimal transport on discrete surfaces

Hugo Lavenant, Sebastian Claici, Edward Chien, Justin Solomon

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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 (finitedimensional) 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.

Original languageEnglish
Article number250
JournalACM Transactions on Graphics
Volume37
Issue number6
DOIs
StatePublished - Nov 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 Copyright held by the owner/author(s).

Funding

J. Solomon acknowledges the generous support of Army Research Office grant W911NF-12-R-0011 ("Smooth Modeling of Flows on Graphs"), of National Science Foundation grant IIS-1838071 ("BIG-DATA:F: Statistical and Computational Optimal Transport for Geometric Data Analysis"), from the MIT Research Support Committee, from an Amazon Research Award, from the MIT-IBM Watson AI Laboratory, and from the Skoltech-MI Next Generation ProgramT. Most of this work was done during a visit of H. Lavenant to MIT; the hospitality of CSAIL and MIT is warmly acknowledged. The authors thank the reviewers for their helpful feedback. J. Solomon acknowledges the generous support of Army Research Office grant W911NF-12-R-0011 (“Smooth Modeling of Flows on Graphs”), of National Science Foundation grant IIS-1838071 (“BIG-DATA:F: Statistical and Computational Optimal Transport for Geometric Data Analysis”), from the MIT Research Support Committee, from an Amazon Research Award, from the MIT–IBM Watson AI Laboratory, and from the Skoltech–MI Next Generation ProgramT. Most of this work was done during a visit of H. Lavenant to MIT; the hospitality of CSAIL and MIT is warmly acknowledged. The authors thank the reviewers for their helpful feedback.

FundersFunder number
IBM Watson AI Laboratory
MIT Research Support Committee
Skoltech–MI Next Generation ProgramT
National Science FoundationIIS-1838071
Army Research OfficeW911NF-12-R-0011
Massachusetts Institute of Technology
National Science Foundation

    Keywords

    • Discrete differential geometry
    • Optimal transport
    • Wasserstein distance

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