Causal Optimal Coupling for Gaussian Input-Output Distributional Data

• URL: http://arxiv.org/abs/2604.01406v1
• Date: Wed, 01 Apr 2026 21:12:51 GMT
• Title: Causal Optimal Coupling for Gaussian Input-Output Distributional Data
• Abstract summary: We study the problem of identifying an optimal coupling between input-output distributional data generated by a causal system.<n>We formulate this problem as a Schr"odinger Bridge, which seeks the coupling closest - in Kullback-Leibler divergence - to a given prior.
• Score: 3.652509571098291
• License: http://creativecommons.org/licenses/by/4.0/
• 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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