# Relative Optimal Transport

arXiv: https://arxiv.org/abs/2411.05678 · HTML: https://mlsanity.com/paper/2411.05678/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2024.

## Built on by (most cited first, 1 of 1)

- [You Need Reasoning to Learn Reasoning: The Limitations of Label-Free RL in Weak Base Models](https://mlsanity.com/paper/2511.04902/) (2025)

## Abstract (arXiv)

We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest.