# The Directional Optimal Transport

arXiv: https://arxiv.org/abs/2002.08717 · HTML: https://mlsanity.com/paper/2002.08717/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2020.

## Built on by (most cited first, 2 of 2)

- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transport](https://mlsanity.com/paper/2606.04695/) (2026)

## Abstract (arXiv)

We introduce a constrained optimal transport problem where origins x can only be transported to destinations y≥ x. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect Y-X given marginals when the effect is monotone, or Y≥ X. We thus focus on supermodular costs (or submodular rewards) and introduce a coupling P* that is optimal for all such costs and yields the sharp bound. This coupling admits manifold characterizations -- geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel -- that explain its structure and imply useful bounds. When the first marginal is atomless, P* is concentrated on the graphs of two maps which can be described in terms of the marginals, the second map arising due to the binding constraint.