AI 中文总结
研究马尔可夫过程优化问题，针对维度诅咒，基于顺序耦合、局部边缘和聚类矩开发凸松弛，提供下界并恢复低阶统计，还将动态最优传输作为特例，开发从松弛解恢复动力学的程序并扩展到一般马尔可夫过程。
AI 中文摘要
本文研究在两个给定概率分布之间进行插值同时最小化给定成本的马尔可夫过程优化问题。主要计算挑战是维度诅咒。为解决此问题，我们根据顺序耦合重新表述问题，并基于局部边缘和聚类矩开发凸松弛。这些松弛利用局部性和稀疏交互结构，提供可计算的下界并恢复中间定律的低阶统计。我们将动态最优传输识别为马尔可夫过程优化问题的特殊情况，并开发从松弛解恢复潜在贝纳穆 - 布雷尼尔动力学的程序。还表明该程序可扩展到更一般的马尔可夫过程，并通过伊辛模型之间的约束过程进行说明。
英文摘要
In this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou--Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.
Comments38 pages, 10 figures
