The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another. It has been well studied and applied to a wide variety of tasks. Here we focus on the dynamic formulation of OT, also known as the Schr\"odinger bridge (SB) problem, which has recently seen a surge of interest in machine learning due to its connections with diffusion-based generative models. In contrast to the static setting, much less is known about the dynamic setting, even for Gaussian distributions. In this paper, we provide closed-form expressions for SBs between Gaussian measures. In contrast to the static Gaussian OT problem, which can be simply reduced to studying convex programs, our framework for solving SBs requires significantly more involved tools such as Riemannian geometry and generator theory. Notably, we establish that the solutions of SBs between Gaussian measures are themselves Gaussian processes with explicit mean and covariance kernels, and thus are readily amenable for many downstream applications such as generative modeling or interpolation. To demonstrate the utility, we devise a new method for modeling the evolution of single-cell genomics data and report significantly improved numerical stability compared to existing SB-based approaches.

翻译：静态最优传输（OT）问题旨在恢复高斯分布间的最优映射（或更一般地，一种耦合）以实现形态转换，该问题已被广泛研究并应用于多种任务。本文聚焦于最优传输的动态形式——薛定谔桥（SB）问题，该问题因与基于扩散的生成模型之间的关联，近年来在机器学习领域引起了广泛关注。与静态情形不同，动态情形（即使针对高斯分布）的研究尚不充分。本文为高斯测度之间的薛定谔桥提供了闭式表达式。与可简化为凸规划问题的静态高斯OT问题形成对比，我们求解SB的框架需要更复杂的工具（如黎曼几何与生成子理论）。值得注意的是，我们证明了高斯测度间SB的解自身即为具有显式均值与协方差核的高斯过程，因此可便捷地应用于生成建模或插值等下游任务。为展示其应用价值，我们设计了一种单细胞基因组学数据演化建模的新方法，并与现有基于SB的方法相比，其数值稳定性显著提升。