提出一种非参数化方法求解薛定谔桥问题，可高效逼近势函数并保证收敛性。
Forward Reverse Kernel Regression for the Schrödinger bridge problem
- 基于核回归与皮卡迭代，构建前后向随机模拟算法
- 证明了势函数估计的收敛速率且达到最优
- 适用于高维扩散过程的条件分布采样，无需嵌套蒙特卡洛
本文研究薛定谔桥问题（SBP），这是熵正则最优传输的核心问题。针对一般参考过程和始末分布，提出一种前向-后向迭代蒙特卡洛方法，非参数化地逼近薛定谔势函数。特别地，在皮卡迭代对应的不动点问题框架下，采用基于核的蒙特卡洛回归。通过在迭代中保持正值性和希尔伯特度量下的压缩性，设计出可证明收敛的算法，并给出势函数估计的收敛速率，且证明其最优性。最后，作为应用，提出一种非嵌套蒙特卡洛方法，用于计算薛定谔桥过程的终态分布，该方法基于构造的势函数及条件扩散的前向-后向模拟技术。
原文摘要 · Abstract (English)
In this paper, we study the Schrödinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin--endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure to approximate the Schrödinger potentials in a nonparametric way. In particular, we use kernel based Monte Carlo regression in the context of Picard iteration of a corresponding fixed point problem. By preserving in the iteration positivity and contractivity in a Hilbert metric sense, we develop a provably convergent algorithm. Furthermore, we provide convergence rates for the potential estimates and prove their optimality. Finally, as an application, we propose a non-nested Monte Carlo procedure for the final dimensional distributions of the Schrödinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions.
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