"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.
"Computational Optimal Transport" is a book that talks about a math concept called Optimal Transport (OT) and how it can be used in data sciences. OT is a powerful tool that helps us compare different sets of numbers. It was first studied by Monge in the 18th century and later developed by other mathematicians. In the 1940s, people started using computers to solve OT problems, and in the 1990s, they found new ways to use it. Now, OT is used in fields like imaging sciences, computer vision, and machine learning. "Computational Optimal Transport" focuses on how to solve OT problems using computers and why it's important for data sciences.
Exploring the Mathematical Concept of Optimal Transport
Optimal transport (OT) is a powerful tool at the intersection of probability, analysis, and optimization that has been used for centuries to compare probability distributions. The concept was first introduced by Monge in the 18th century and 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 OT numerically using linear programming techniques in 1949 and throughout the following decade. This breakthrough paved the way for significant industrial applications throughout the second half of the 20th century.
In recent years, 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: A Comprehensive Review
"Computational Optimal Transport" is a book written by Gabriel Peyré and Marco Cuturi that explores these developments in detail. It focuses on numerical methods for solving OT problems as well as their practical applications in data sciences – providing readers with a deeper understanding of both its mathematical foundations as well as its real-world implications.
The book begins with an introduction to OT's theoretical properties which make it particularly useful for data science applications such as imaging sciences (e.g., color or texture processing), computer vision/graphics (for shape manipulation), machine learning (for regression, classification, etc.), etc..
Overall "Computational Optimal Transport" provides readers with an insightful overview into this fascinating topic – from its historical origins through modern day developments – all presented through contributions from two experts who have dedicated much time exploring this field: Gabriel Peyré & Marco Cuturi
Optimal Transport (OT) is a mathematical gem at the interface between probability, analysis and optimization. The goal of that theory is to define geometric tools that are useful to compare probability distributions. Earlier contributions originated from Monge's work in the 18th century, to be later rediscovered under a different formalism by Tolstoi in the 1920's, Kantorovich, Hitchcock and Koopmans in the 1940's. The problem was solved numerically by Dantzig in 1949 and others in the 1950's within the framework of linear programming, paving the way for major industrial applications in the second half of the 20th century. OT was later rediscovered under a different light by analysts in the 90's, following important work by Brenier and others, as well as in the computer vision/graphics fields under the name of earth mover's distances. Recent years have witnessed yet another revolution in the spread of OT, 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.
