Notions of optimal transport theory and how to implement them on a computer

TL;DR

This paper introduces optimal transport theory, explaining its core principles and notions, and demonstrates how to implement it computationally, especially in the semi-discrete setting using Laguerre diagrams for efficient algorithms.

Contribution

It provides an accessible explanation of optimal transport theory and details an efficient computational approach for the semi-discrete case using Laguerre diagrams.

Findings

Introduction of optimal transport concepts and their relations

Development of an efficient algorithm for semi-discrete optimal transport

Application of computational geometry tools like Laguerre diagrams

Abstract

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.