We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold ℳ d embedded in ℝ p , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold.

中文翻译：

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矢量扩散图和拉普拉斯连接


我们引入矢量扩散图（VDM），这是一种用于组织和分析大量高维数据集、图像和形状的新数学框架。 VDM 是扩散图和其他非线性降维方法（例如 LLE、ISOMAP 和拉普拉斯特征图）的数学和算法推广。虽然现有方法直接或间接与数据函数的热核相关，但 VDM 是基于矢量场的热核。 VDM 提供了用于组织复杂数据集、将其嵌入低维空间以及对数据进行插值和回归向量场的工具。特别是，它为数据配备了一个度量，我们将其称为向量扩散距离。在流形学习设置中，数据集分布在嵌入 ℝ p 的低维流形 ℳ d 上，我们证明了 VDM 与流形上向量场的连接拉普拉斯算子之间的关系。