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 language | English |
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Article number | 250 |
Journal | ACM Transactions on Graphics |
Volume | 37 |
Issue number | 6 |
DOIs | |
State | Published - Nov 2018 |
Externally published | Yes |
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.
Funders | Funder number |
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IBM Watson AI Laboratory | |
MIT Research Support Committee | |
Skoltech–MI Next Generation ProgramT | |
National Science Foundation | IIS-1838071 |
Army Research Office | W911NF-12-R-0011 |
Massachusetts Institute of Technology | |
National Science Foundation |
Keywords
- Discrete differential geometry
- Optimal transport
- Wasserstein distance