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.
Original language | English |
---|---|
Title of host publication | SIGGRAPH Asia 2018 Technical Papers, SIGGRAPH Asia 2018 |
Publisher | Association for Computing Machinery, Inc |
ISBN (Electronic) | 9781450360081 |
DOIs | |
State | Published - 4 Dec 2018 |
Externally published | Yes |
Event | SIGGRAPH Asia 2018 Technical Papers - International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH Asia 2018 - Tokyo, Japan Duration: 4 Dec 2018 → 7 Dec 2018 |
Publication series
Name | SIGGRAPH Asia 2018 Technical Papers, SIGGRAPH Asia 2018 |
---|
Conference
Conference | SIGGRAPH Asia 2018 Technical Papers - International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH Asia 2018 |
---|---|
Country/Territory | Japan |
City | Tokyo |
Period | 4/12/18 → 7/12/18 |
Bibliographical note
Publisher Copyright:© 2018 Copyright held by the owner/author(s). Publication rights licensed to ACM.
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.
Funders | Funder number |
---|---|
Fundação Amazônia Paraense de Amparo à Pesquisa | |
University Research Committee, University of Hong Kong | |
MIT Research Support Committee | |
Skolkovo Institute of Science and Technology | |
National Science Foundation | |
Lincoln Laboratory, Massachusetts Institute of Technology | |
IBM Watson AI Laboratory | |
Massachusetts Institute of Technology | |
Skoltech–MI Next Generation ProgramT | |
National Science Foundation | 1838071 |
Army Research Office | W911NF-12-R-0011 |
Directorate for Computer and Information Science and Engineering | 1838071 |
Keywords
- Discrete Differential Geometry
- Optimal Transport
- Wasserstein Distance