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April 2022 The directional optimal transport
Marcel Nutz, Ruodu Wang
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Ann. Appl. Probab. 32(2): 1400-1420 (April 2022). DOI: 10.1214/21-AAP1712


We introduce a constrained optimal transport problem where origins x can only be transported to destinations yx. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect YX given marginals when the effect is monotone, or YX. 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.

Funding Statement

The first author was supported by an Alfred P. Sloan Fellowship and NSF Grant DMS-1812661.
The second author was supported by NSERC Grants RGPIN-2018-03823 and RGPAS-2018-522590.


The authors are indebted to Mathias Beiglböck, Filippo Santambrogio and Julian Schuessler for fruitful discussions that greatly helped this work. The authors also thank the review team for helpful comments.


Download Citation

Marcel Nutz. Ruodu Wang. "The directional optimal transport." Ann. Appl. Probab. 32 (2) 1400 - 1420, April 2022.


Received: 1 February 2020; Revised: 1 March 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1712

Primary: 49N05 , 62G10 , 93E20

Keywords: monotone treatment effect , Optimal transport , submodular reward

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 2 • April 2022
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