Semi-discrete optimal transport

TL;DR

This book introduces semi-discrete optimal transport, emphasizing accessible explanations and applications in information and game theory, while connecting it to classical problems like stable marriage.

Contribution

It presents a self-contained, accessible approach to semi-discrete optimal transport, linking it to classical problems and applications in various fields.

Findings

Optimal transport can be viewed as a special case of the stable marriage problem.

The book connects semi-discrete transport to classical partition and coupling problems.

Applications to information theory and game theory are explored.

Abstract

In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete cases, and always assume compactness for the underlying spaces. However, some fundamental knowledge of measure theory and convexity is unavoidable. In order to make it as self-contained as possible I included an appendix with some basic definitions and results. I believe that any graduate student in mathematics, as well as advanced undergraduate students, can read and understand this book. Some chapters (in particular in Parts II\&III ) can also be interesting for experts. Starting with the the most fundamental, fully discrete problem I attempted to place optimal transport as a particular case of the celebrated stable marriage problem. From there we proceed to the partition problem, which can be formulated as a transport from a continuous space to a discrete one. Applications to information theory and game theory (cooperative and non-cooperative) are introduced as well. Finally, the general case of transport between two compact measure spaces is introduced as a coupling between two semi-discrete transports.