Markov Kernels in Optimal Transport via Extending c-Cyclic Monotonicity

TL;DR

This paper introduces a novel framework using Markov kernels for optimal transport, enabling extensions to signed measures and revealing new structural insights, especially in one-dimensional cases.

Contribution

It presents a new approach that models optimal transport via Markov kernels, extending the theory to signed measures and providing structural insights.

Findings

Markov kernels can model optimal transport problems.

The framework extends to signed measures.

Additional structure is revealed in one-dimensional signed optimal transport.

Abstract

In this paper we show that we can use Markov kernels as a model for optimal transport. This new framework can be easily translated into the standard coupling formulation of optimal transport, and we show that we can use a coupling as a Markov kernel for many optimal transport problems. Using kernels allows us to extend optimal transport to signed measures and treats the support of the measure as the salient feature. This approach reveals additional structure for one-dimensional signed optimal transport.