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Computational Optimal Transport

Gabriel Peyré, Marco Cuturi

TL;DR

This document serves as a practical guide for producing publication-ready papers using the now LaTeX class, with a focus on converting existing LaTeX sources, structuring frontmatter and backmatter, and adhering to journal-specific style guidelines. It covers file preparation, conversion steps, typography and formatting rules, and the organization of content (including theorems, figures, and references) to ensure consistency across formats. It also provides a hands-on set of examples and a minimal skeleton of the document to illustrate proper usage, along with final-stage fine-tuning instructions. The practical impact is to streamline the creation of polished, publication-ready documents for Foundations and Trends and similar journals, reducing formatting friction and improving cross-format compatibility.

Abstract

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

TL;DR

This document serves as a practical guide for producing publication-ready papers using the now LaTeX class, with a focus on converting existing LaTeX sources, structuring frontmatter and backmatter, and adhering to journal-specific style guidelines. It covers file preparation, conversion steps, typography and formatting rules, and the organization of content (including theorems, figures, and references) to ensure consistency across formats. It also provides a hands-on set of examples and a minimal skeleton of the document to illustrate proper usage, along with final-stage fine-tuning instructions. The practical impact is to streamline the creation of polished, publication-ready documents for Foundations and Trends and similar journals, reducing formatting friction and improving cross-format compatibility.

Abstract

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Paper Structure

This paper contains 22 sections, 4 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Obtaining the required files
  3. Preparing the LaTeX source
  4. Converting Your LaTeX Source
  5. Style Guidelines
  6. Bibliography
  7. Referring to your paper
  8. Titles
  9. References
  10. Preface and other special chapters
  11. Figures and plots
  12. Mathematics
  13. Source code
  14. Making the output look good
  15. Bare Bones File Outline
  16. ...and 7 more sections

Key Result

Theorem B.1

Let $(X, d)$ be a non-empty complete metric space with a contraction mapping $T : X \to X$. Then $T$ admits a unique fixed-point $x^*$ in $X$ (i.e., $T(x^*) = x^*$). Furthermore, $x^*$ can be found as follows: start with an arbitrary element $x_0$ in X and define a sequence $x_n$ by $x_n = T(x_{n-1}

Theorems & Definitions (7)

  • Definition B.1: contraction mapping
  • Theorem B.1: Banach fixed point theorem
  • Lemma B.2
  • Corollary B.3
  • Remark B.1
  • Proposition B.1
  • Example B.1