提出因果最优耦合方法，用于建模动态系统的输入输出分布关系。
Causal Optimal Coupling for Gaussian Input-Output Distributional Data
- 将因果系统数据耦合建模为施罗丁格桥问题，融合边际分布与时序约束
- 对高斯分布与二次代价函数情形，给出可计算的迭代求解方案
- 为分布数据下的系统辨识提供理论基础，适合因果推理与控制研究者
我们研究由因果动态系统生成的输入输出分布数据之间的最优耦合识别问题。耦合需满足给定的边缘分布及反映系统时间结构的因果性约束。该问题被形式化为一个施罗丁格桥问题，即在相对熵意义下寻找最接近给定先验的耦合，同时满足边缘分布和因果性约束。对于高斯边缘分布与一般时变二次代价函数的情形，我们推导出可完全解析的Sinkhorn迭代表征，其收敛至最优解。除理论贡献外，该框架为从分布数据中应用因果最优传输方法进行系统辨识提供了原则性基础。
原文摘要 · Abstract (English)
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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