The Schr\"odinger bridge problem (SBP) is gaining increasing attention in
generative modeling and showing promising potential even in comparison with the
score-based generative models (SGMs). SBP can be interpreted as an
entropy-regularized optimal transport problem, which conducts projections onto
every other marginal alternatingly. However, in practice, only approximated
projections are accessible and their convergence is not well understood. To
fill this gap, we present a first convergence analysis of the Schr\"odinger
bridge algorithm based on approximated projections. As for its practical
applications, we apply SBP to probabilistic time series imputation by
generating missing values conditioned on observed data. We show that optimizing
the transport cost improves the performance and the proposed algorithm achieves
the state-of-the-art result in healthcare and environmental data while
exhibiting the advantage of exploring both temporal and feature patterns in
probabilistic time series imputation.

本文提出了基于逼近投影的 Schrödinger bridge 算法的首个收敛性分析，成功地将其应用于概率时间序列插补中，以生成缺失值并达到了业内的最佳表现。通过优化传输成本，该算法探索了概率时间序列中的时间和特征模式。

可证收敛的 Schrödinger 桥及其在概率时间序列插值中的应用

Provably Convergent Schrödinger Bridge with Applications to  Probabilistic Time Series Imputation

Schr\"odinger Bridge (SB) is an entropy-regularized optimal transport problem
that has received increasing attention in deep generative modeling for its
mathematical flexibility compared to the Scored-based Generative Model (SGM).
However, it remains unclear whether the optimization principle of SB relates to
the modern training of deep generative models, which often rely on constructing
log-likelihood objectives.This raises questions on the suitability of SB models
as a principled alternative for generative applications. In this work, we
present a novel computational framework for likelihood training of SB models
grounded on Forward-Backward Stochastic Differential Equations Theory - a
mathematical methodology appeared in stochastic optimal control that transforms
the optimality condition of SB into a set of SDEs. Crucially, these SDEs can be
used to construct the likelihood objectives for SB that, surprisingly,
generalizes the ones for SGM as special cases. This leads to a new optimization
principle that inherits the same SB optimality yet without losing applications
of modern generative training techniques, and we show that the resulting
training algorithm achieves comparable results on generating realistic images
on MNIST, CelebA, and CIFAR10. Our code is available at
this https URL

本文提出了一种新的计算框架，基于正向 - 反向随机微分方程理论，用于似然训练 Schr"odinger Bridge 模型，并展示了该优化原则具有与深度生成模型现代训练技术相适应的适用性，实现了对 MNIST、CelebA 和 CIFAR10 等数据集生成逼真图像的可比较结果。