Causal Optimal Coupling for Gaussian Input-Output Distributional Data

TL;DR

This paper introduces a causal optimal transport framework for Gaussian distributional data, formulating it as a Schr"odinger Bridge problem with a focus on causality and marginal constraints.

Contribution

It develops a tractable characterization of Sinkhorn iterations for Gaussian cases, enabling practical causal system identification from distributional data.

Findings

Derived a convergent Sinkhorn iteration for Gaussian causal optimal transport.

Formulated the problem as a Schr"odinger Bridge with causality constraints.

Provided a theoretical foundation for causal system identification using distributional data.

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.