将最优传输与卡尔曼滤波结合，实现高斯分布间高效精准计算
Gaussian entropic optimal transport: Schrödinger bridges and the Sinkhorn algorithm
- 基于递归公式构建高斯分布的迭代比例调整算法
- 首次给出高斯熵最优传输的闭式解及传输映射表达式
- 适用于生成模型与概率推断，尤其适合多维高斯场景
熵正则化最优传输问题在机器学习与生成建模中日益重要。对于有限空间，通常使用Sinkhorn算法求解。但在更一般的设定下，传统迭代方法依赖非线性条件/共轭变换，难以获得精确的有限维解。本文提出一种针对一般高斯多元模型的有限维递归形式的迭代比例调整过程。该方法与著名的卡尔曼滤波及其相关的Riccati矩阵差分方程密切相关，可直接在实际场景中实施而无需额外近似。我们进一步扩展该滤波框架，提供高斯版Sinkhorn算法的完整收敛性分析，并推导出熵最优传输映射和Schrödinger桥的闭式表达式。
原文摘要 · Abstract (English)
Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finite-dimensional solutions cannot be computed. This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges.
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