# Causal Optimal Coupling for Gaussian Input-Output Distributional Data

arXiv: https://arxiv.org/abs/2604.01406 · HTML: https://mlsanity.com/paper/2604.01406/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 14 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 7 of 7)

- [Causal transport in discrete time and applications](https://mlsanity.com/paper/1606.04062/) (2016)
- [Computational methods for adapted optimal transport](https://mlsanity.com/paper/2203.05005/) (2022)
- [Dynamic Programming in Probability Spaces via Optimal Transport](https://mlsanity.com/paper/2302.13550/) (2023)
- [A convex approach for Markov chain estimation from aggregate data via inverse optimal transport](https://mlsanity.com/paper/2511.16458/) (2025)
- [Consistently Estimating Markov Chains with Noisy Aggregate Data](https://mlsanity.com/paper/1604.04182/) (2016)
- [Entropic adapted Wasserstein distance on Gaussians](https://mlsanity.com/paper/2412.18794/) (2024)
- [Causal Tracking of Distributions in Wasserstein Space: A Model Predictive Control Scheme](https://mlsanity.com/paper/2403.15702/) (2024)

## Abstract (arXiv)

We study the problem of identifying an optimal coupling between input-output distributional data generated by a causal dynamical system. The coupling is required to satisfy prescribed marginal distributions and a causality constraint reflecting the temporal structure of the system. We formulate this problem as a Schr"odinger Bridge, which seeks the coupling closest - in Kullback-Leibler divergence - to a given prior while enforcing both marginal and causality constraints. For the case of Gaussian marginals and general time-dependent quadratic cost functions, we derive a fully tractable characterization of the Sinkhorn iterations that converges to the optimal solution. Beyond its theoretical contribution, the proposed framework provides a principled foundation for applying causal optimal transport methods to system identification from distributional data.