Wilson Michael, Needham Tom, Park Chiwoo, Kundu Suparteek, Srivastava Anuj
Department of Statistics, Florida State University, Tallahassee, FL 32306 USA.
Department of Mathematics, Florida State University, Tallahassee, FL 32306 USA.
SIAM J Imaging Sci. 2024;17(3):1433-1466. doi: 10.1137/23m1620363. Epub 2024 Jul 11.
This paper uses sample data to study the problem of comparing populations on finite-dimensional parallelizable Riemannian manifolds and more general trivial vector bundles. Utilizing triviality, our framework represents populations as mixtures of Gaussians on vector bundles and estimates the population parameters using a mode-based clustering algorithm. We derive a Wasserstein-type metric between Gaussian mixtures, adapted to the manifold geometry, in order to compare estimated distributions. Our contributions include an identifiability result for Gaussian mixtures on manifold domains and a convenient characterization of optimal couplings of Gaussian mixtures under the derived metric. We demonstrate these tools on some example domains, including the preshape space of planar closed curves, with applications to the shape space of triangles and populations of nanoparticles. In the nanoparticle application, we consider a sequence of populations of particle shapes arising from a manufacturing process and utilize the Wasserstein-type distance to perform change-point detection.
本文使用样本数据研究有限维可并行化黎曼流形以及更一般的平凡向量丛上的总体比较问题。利用平凡性，我们的框架将总体表示为向量丛上高斯分布的混合，并使用基于模式的聚类算法估计总体参数。为了比较估计的分布，我们推导了一种适用于流形几何的高斯混合之间的瓦瑟斯坦型度量。我们的贡献包括流形域上高斯混合的可识别性结果，以及在推导度量下高斯混合最优耦合的便捷特征。我们在一些示例域上展示了这些工具，包括平面封闭曲线的预形状空间，并将其应用于三角形的形状空间和纳米颗粒总体。在纳米颗粒应用中，我们考虑由制造过程产生的一系列颗粒形状总体，并利用瓦瑟斯坦型距离进行变点检测。
