# Causal transport in discrete time and applications

arXiv: https://arxiv.org/abs/1606.04062 · HTML: https://mlsanity.com/paper/1606.04062/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2016.

## Built on by (most cited first, 21 of 21)

- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Weisfeiler-Lehman meets Gromov-Wasserstein](https://mlsanity.com/paper/2202.02495/) (2022)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Time-Causal VAE: Robust Financial Time Series Generator](https://mlsanity.com/paper/2411.02947/) (2024)
- [The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs](https://mlsanity.com/paper/2302.00713/) (2023)
- [Bisimulation Metrics are Optimal Transport Distances, and Can be Computed Efficiently](https://mlsanity.com/paper/2406.04056/) (2024)
- [Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairness](https://mlsanity.com/paper/2408.03425/) (2024)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Conditional COT-GAN for Video Prediction with Kernel Smoothing](https://mlsanity.com/paper/2106.05658/) (2021)
- [Fitted value iteration methods for bicausal optimal transport](https://mlsanity.com/paper/2306.12658/) (2023)
- [Distances for Markov chains from sample streams](https://mlsanity.com/paper/2505.18005/) (2025)
- [Distributionally robust risk evaluation with a causality constraint and structural information](https://mlsanity.com/paper/2203.10571/) (2022)
- [Auditing and Enforcing Conditional Fairness via Optimal Transport](https://mlsanity.com/paper/2410.14029/) (2024)
- [Nested Optimal Transport Distances](https://mlsanity.com/paper/2509.06702/) (2025)
- [What is a good matching of probability measures? A counterfactual lens on transport maps](https://mlsanity.com/paper/2509.16027/) (2025)
- [New Algorithmic Directions in Optimal Transport and Applications for Product Spaces](https://mlsanity.com/paper/2509.21502/) (2025)
- [Wasserstein Distributionally Robust Optimization Through the Lens of Structural Causal Models and Individual Fairness](https://mlsanity.com/paper/2509.26275/) (2025)
- [Designing Ambiguity Sets for Distributionally Robust Optimization Using Structural Causal Optimal Transport](https://mlsanity.com/paper/2510.00599/) (2025)
- [Causal Optimal Coupling for Gaussian Input-Output Distributional Data](https://mlsanity.com/paper/2604.01406/) (2026)
- [From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transport](https://mlsanity.com/paper/2606.04695/) (2026)
- [Gumbel Straight Flow: Distilling Autoregressive Models into One-step Flow Maps](https://mlsanity.com/paper/2610.00497/) (2026)

## Abstract (arXiv)

Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier's map. Moreover, these considerations provide transport-information inequalities for the nested distance between stochastic processes pioneered by Pflug and Pichler, and so serve to gauge the discrepancy between stochastic programs driven by different noise distributions.