# The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation

## TL;DR

The paper introduces the boundary method, a new technique for efficiently solving semi-discrete optimal transport problems across various cost functions, with theoretical backing and practical testing.

## Contribution

It presents the boundary method that reduces problem complexity and provides convergence analysis for p-norm cost functions, extending applicability to general costs.

## Key findings

- Effective reduction in problem dimension
- Convergence proven for p-norm costs
- Successful testing on various cost functions

## Abstract

We introduce a new technique, which we call the boundary method, for solving semi-discrete optimal transport problems with a wide range of cost functions. The boundary method reduces the effective dimension of the problem, thus improving complexity. For cost functions equal to a p-norm with p in (1,infinity), we provide mathematical justification, convergence analysis, and algorithmic development. Our testing supports the boundary method with these p-norms, as well as other, more general cost functions.