在概率分布空间中定义了布塞曼函数，实现对高斯混合模型的高效投影与距离计算。
Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing
- 提出在沃尔泰拉空间中计算布塞曼函数的闭式解，适用于一维分布和高斯分布。
- 基于闭式解构建新型切片沃瑟斯坦距离，支持对高斯混合模型的显式投影。
- 方法在合成数据与迁移学习任务中表现高效，适合几何机器学习研究者。
布塞曼函数在几何机器学习中日益受到关注，因其自然定义了流形上测地射线的投影，并推广了超平面概念。由于多种数据可建模为概率分布，研究该函数在沃尔泰拉空间中的存在性与计算具有重要意义，该空间由最优传输度量诱导出丰富的形式黎曼结构。本文研究了沃尔泰拉空间中布塞曼函数的存在性与计算问题，其允许测地射线。我们建立了两种重要情形下的闭式表达：一维分布和高斯分布。这些结果使得在实数轴上的概率分布投影成为可能，进而支持对高斯混合模型及标注数据集定义新型切片沃瑟斯坦距离。我们在合成数据集和迁移学习任务中验证了这些新方案的有效性。
原文摘要 · Abstract (English)
The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。