# Computational methods for adapted optimal transport

arXiv: https://arxiv.org/abs/2203.05005 · HTML: https://mlsanity.com/paper/2203.05005/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2022.

## Built on by (most cited first, 12 of 12)

- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (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)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [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)
- [Nested Optimal Transport Distances](https://mlsanity.com/paper/2509.06702/) (2025)
- [Wasserstein Distributionally Robust Optimization Through the Lens of Structural Causal Models and Individual Fairness](https://mlsanity.com/paper/2509.26275/) (2025)
- [Robust Optimization in Causal Models and G-Causal Normalizing Flows](https://mlsanity.com/paper/2510.15458/) (2025)
- [Causal Optimal Coupling for Gaussian Input-Output Distributional Data](https://mlsanity.com/paper/2604.01406/) (2026)

## Abstract (arXiv)

Adapted optimal transport (AOT) problems are optimal transport problems for distributions of a time series where couplings are constrained to have a temporal causal structure. In this paper, we develop computational tools for solving AOT problems numerically. First, we show that AOT problems are stable with respect to perturbations in the marginals and thus arbitrary AOT problems can be approximated by sequences of linear programs. We further study entropic methods to solve AOT problems. We show that any entropically regularized AOT problem converges to the corresponding unregularized problem if the regularization parameter goes to zero. The proof is based on a novel method - even in the non-adapted case - to easily obtain smooth approximations of a given coupling with fixed marginals. Finally, we show tractability of the adapted version of Sinkhorn's algorithm. We give explicit solutions for the occurring projections and prove that the procedure converges to the optimizer of the entropic AOT problem.