Causal Optimal Coupling for Gaussian Input-Output Distributional Data

2026-04-01

AI Summary

This work addresses the problem of identifying the optimal coupling between input and output distributions generated by a causal dynamical system, subject to both a prescribed causal temporal structure and given Gaussian marginals. The authors formulate this as a Schrödinger bridge problem with explicit causality constraints, minimizing the Kullback–Leibler divergence to a prior coupling under a time-varying quadratic cost function. Their key contribution is the derivation of a closed-form Sinkhorn-type iterative algorithm for the Gaussian setting, enabling efficient and analytically tractable solutions to the causal optimal transport problem. The proposed method rigorously preserves causality and temporal consistency while offering a novel theoretical framework and computational paradigm for distributional data-driven system identification.

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.

Problem

• causal optimal coupling
• Gaussian distributions
• Schrödinger Bridge
• causality constraint
• distributional data

Innovation

• Causal Optimal Transport
• Schrödinger Bridge
• Gaussian Distributions
• Sinkhorn Iterations
• System Identification