# What is a good matching of probability measures? A counterfactual lens on transport maps

arXiv: https://arxiv.org/abs/2509.16027 · HTML: https://mlsanity.com/paper/2509.16027/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 78 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 30 of 39)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Theoretical Analysis of Domain Adaptation with Optimal Transport](https://mlsanity.com/paper/1610.04420/) (2016)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Counterfactual Credit Assignment in Model-Free Reinforcement Learning](https://mlsanity.com/paper/2011.09464/) (2020)
- [Unified Optimal Transport Framework for Universal Domain Adaptation](https://mlsanity.com/paper/2210.17067/) (2022)
- [FlipTest: Fairness Testing via Optimal Transport](https://mlsanity.com/paper/1906.09218/) (2019)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Fair Data Adaptation with Quantile Preservation](https://mlsanity.com/paper/1911.06685/) (2019)
- [Personalized Decision Making -- A Conceptual Introduction](https://mlsanity.com/paper/2208.09558/) (2022)
- [Counterfactual harm](https://mlsanity.com/paper/2204.12993/) (2022)
- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [Causal transport in discrete time and applications](https://mlsanity.com/paper/1606.04062/) (2016)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [Optimal transport and Wasserstein distances for causal models](https://mlsanity.com/paper/2303.14085/) (2023)
- [An Optimal Transport Approach to Estimating Causal Effects via Nonlinear Difference-in-Differences](https://mlsanity.com/paper/2108.05858/) (2021)
- [Counterfactual Cocycles: A Framework for Robust and Coherent Counterfactual Transports](https://mlsanity.com/paper/2405.13844/) (2024)

## Abstract (arXiv)

Coupling probability measures lies at the core of many problems in statistics and machine learning, from domain adaptation to transfer learning and causal inference. Yet, even when restricted to deterministic transports, such couplings are not identifiable: two atomless marginals admit infinitely many transport maps. The common recourse to optimal transport, motivated by cost minimization and cyclical monotonicity, obscures the fact that several distinct notions of multivariate monotone matchings coexist. In this work, we first carry a comparative analysis of three constructions of transport maps: cyclically monotone, quantile-preserving and triangular monotone maps. We establish necessary and sufficient conditions for their equivalence, thereby clarifying their respective structural properties. In parallel, we formulate counterfactual reasoning within the framework of structural causal models as a problem of selecting transport maps between fixed marginals, which makes explicit the role of untestable assumptions in counterfactual reasoning. Then, we are able to connect these two perspectives by identifying conditions on causal graphs and structural equations under which counterfactual maps coincide with classical statistical transports. In this way, we delineate the circumstances in which causal assumptions support the use of a specific structure of transport map. Taken together, our results aim to enrich the theoretical understanding of families of transport maps and to clarify their possible causal interpretations. We hope this work contributes to establishing new bridges between statistical transport and causal inference.