Causal Optimal Coupling for Gaussian Input-Output Distributional Data

Daran Xu; Amirhossein Taghvaei

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arXiv · 2604.01406

Causal Optimal Coupling for Gaussian Input-Output Distributional Data

Daran Xu·Amirhossein Taghvaei

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Abstract

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.

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BibTeX RIS

Daran Xu, Amirhossein Taghvaei. 2026-04-01. Causal Optimal Coupling for Gaussian Input-Output Distributional Data. https://arxiv.org/abs/2604.01406

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