# Computational Optimal Transport

**Authors:** Gabriel Peyré, Marco Cuturi  
**arXiv:** [1803.00567](https://arxiv.org/abs/1803.00567)  
**Date:** 2018-03-01  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1803.00567v1/

## Key points

- "Computational Optimal Transport" is a book by Gabriel Peyré and Marco Cuturi
- The book explores the mathematical concept of Optimal Transport (OT) and its applications in data sciences
- OT is a powerful tool at the intersection of probability, analysis, and optimization
- OT aims to define geometric tools for comparing probability distributions
- OT has origins in Monge's work in the 18th century and was later developed by Kantorovich, Hitchcock, Koopmans, and others
- Dantzig and others solved OT numerically using linear programming techniques in the 1940s
- In the 1990s, analysts rediscovered OT from a different perspective with contributions from Brenier and others
- Recent advancements in approximate solvers have made OT scalable for large-scale problems relevant to data sciences
- OT is increasingly used in fields such as imaging sciences, computer vision and graphics, and machine learning
- "Computational Optimal Transport" focuses on numerical methods for solving OT problems and their practical applications in data sciences
- The book also explores theoretical properties of OT that make it useful for these applications

## Summary

"Computational Optimal Transport" is a book written by Gabriel Peyré and Marco Cuturi that explores the mathematical concept of Optimal Transport (OT) and its applications in data sciences. OT is a powerful tool at the intersection of probability, analysis, and optimization, with the aim of defining geometric tools to compare probability distributions. The origins of OT can be traced back to Monge's work in the 18th century, which was later rediscovered by Tolstoi in the 1920s. In the 1940s, Kantorovich, Hitchcock, and Koopmans further developed this theory under a different formalism. Dantzig and others then solved the problem numerically using linear programming techniques in 1949 and the following decade. This breakthrough paved the way for significant industrial applications throughout the second half of the 20th century. In the 1990s, analysts rediscovered OT from a different perspective, building upon important contributions by Brenier and others. More recently, there has been a revolution in OT thanks to approximate solvers that can handle large-scale problems relevant to data sciences. This newfound scalability has led to an increasing use of OT in various fields such as imaging sciences (e.g., color or texture processing), computer vision and graphics (for shape manipulation), and machine learning (for regression, classification, and density fitting). "Computational Optimal Transport" focuses on numerical methods for solving OT problems and their practical applications in data sciences. The book also sheds light on theoretical properties of OT that make it particularly useful for these applications. With contributions from Peyré and Cuturi – experts in this field – this comprehensive review provides readers with a deeper understanding of OT's mathematical foundations as well as its real-world implications.