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Causal Optimal Coupling for Gaussian Input-Output Distributional Data

Daran Xu, Amirhossein Taghvaei

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.

Causal Optimal Coupling for Gaussian Input-Output 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.

Paper Structure

This paper contains 5 sections, 2 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Sinkhorn Iteration in the Gaussian Setting
  4. Numerical Example
  5. Conclusion

Key Result

Theorem 1

If $\mu$ is a non-degenerate Gaussian distribution, and $\inf_{\pi' \in \Pi_c(\mu,*)}\,D_\text{KL}(\pi',\gamma) < \infty$, the solution $\pi$ to the following causal projection problem is also Gaussian. The parameters of the conditional distribution $\pi(u_t|u_{1:t-1},y_{1:t-1})$ are given by The parameters of the conditional distribution $\pi(y_t|u_{1:t},y_{1:t-1})$ are given by where $\black

Figures (1)

  • 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.

Theorems & Definitions (3)

  • Remark 1
  • Theorem 1: Odd iteration
  • Theorem 2: Even iteration