具有状态切换的薛定谔桥的Sinkhorn算法的指数收敛性
Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching
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中文总结 AI 辅助
研究具有状态切换的薛定谔桥问题的Sinkhorn算法收敛性，通过构造等效熵最优传输公式进行Sinkhorn迭代，在紧致性假设下证明其在相对熵中指数收敛，还对部分观测终端设置做了类似分析。
中文摘要 AI 辅助
本文研究了如Zlotchevski和Chen（2025）中所介绍的具有状态切换的薛定谔桥问题的Sinkhorn算法的收敛性。我们考虑一类在混合状态空间\(E = \mathbb{R}^d\times\{1,\ldots,m\}\)上的状态切换随机系统，并通过相关的等效熵最优传输公式构造Sinkhorn迭代。本文的主要结果是在紧致性假设下，Sinkhorn算法在相对熵中指数收敛。我们的证明受到Chiarini、Conforti、Greco和Tamanini（2024）、Eckstein（2025）中最近为证明经典薛定谔问题的指数收敛而发展的论证的启发。我们对部分观测终端设置进行了类似分析，即在终端时刻仅规定连续分量的边际分布，而离散状态未被观测。
英文摘要
This paper studies the convergence of the Sinkhorn algorithm for the Schrödinger bridge problem with regime switching, as introduced in Zlotchevski and Chen (2025). We consider a class of regime-switching stochastic systems on the hybrid state space $E=\mathbb{R}^d\times\{1,\ldots,m\}$, and construct the Sinkhorn iteration through the associated equivalent entropic optimal transport formulation. The main result of this paper is the exponential convergence of the Sinkhorn algorithm in relative entropy under compactness assumptions. Our proofs are inspired by the arguments recently developed for proving exponential convergence of the classical Schrödinger problem in Chiarini, Conforti, Greco and Tamanini (2024), Eckstein (2025). We perform a similar analysis for the partially observed terminal setting, where only the marginal distribution of the continuous component is prescribed at the terminal time, while the discrete regime is unobserved.
