Causal Optimal Coupling for Gaussian Input-Output Distributional Data

Daran Xu; Amirhossein Taghvaei

eess.SYApr 1, 2026

Causal Optimal Coupling for Gaussian Input-Output Distributional Data

Authors: Daran Xu, Amirhossein Taghvaei

Organizations: Department of Applied Mathematics, University of Washington, Seattle, WA · Department of Aeronautics and Astronautics, University of Washington, Seattle, WA

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.

Figures & tables

Figure 1: $\text{Cov}(U_{t},Y_{s}|U_{1:\tau})$ with $\tau=0.25$ . The top panel is a causal case, and the bottom panel is a non-causal case. The magnitude is shown on a symmetric logarithmic scale, preserving both sign and zero.

Optimal Transport

Optimal transport compares and transforms probability distributions by finding low-cost ways to move mass between them.

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Stochastic Optimization Optimal Transport

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Bi-causal optimal transport (OT) is a natural framework for comparing and coupling stochastic processes under nonanticipative information constraints, with important applications in robust finance, sequential uncertainty quantification, and multistage stochastic optimization. In particular, a learned bi-causal coupling naturally serves as a simulator for generating joint sample paths that respect both prescribed marginal laws and the underlying information flow. Its practical use, however, is limited by the computational difficulty of enforcing bi-causal coupling constraints over path space, especially for continuous distributions and long horizons. We develop a scalable stochastic-optimization framework for computing bi-causal OT couplings under general marginals. Our approach introduces a Kullback--Leibler (KL)-penalized relaxation that replaces hard marginal constraints with tractable divergence penalties while preserving the recursive structure of the problem. We establish dynamic programming principles for both the original and relaxed formulations, prove that the relaxed problem converges to the original bi-causal OT problem as the penalty grows, and derive explicit policy-gradient representations for the relaxed objective. Building on these results, we propose a practical policy-gradient algorithm with unbiased mini-batch estimators, variance reduction, and nonasymptotic regret guarantees. Numerical experiments show that the method accurately captures marginal laws and temporal dependence, and performs well in applications including robust subhedging and time series statistical downscaling. These results provide a scalable computational approach to bi-causal OT and broaden its applicability in settings where nonanticipative information constraints are essential.

Abstract

Optimal Transport

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