The diffusion geometry of fibre bundles: Horizontal diffusion maps
The diffusion geometry of fibre bundles: Horizontal diffusion maps
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DOI：
10.1016/j.acha.2019.08.001
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发表时间：
2016-02
影响因子：
2.5
通讯作者：
Tingran Gao
中科院分区：
文献类型：
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作者：
Tingran Gao
基于核的非线性降维方法，如局部线性嵌入(LLE)和拉普拉斯特征映射，在很大程度上依赖于成对距离或相似性得分，利用它们可以构造和研究与数据集相关的加权图。然而，当每个单独的数据对象携带额外的结构细节时，这些结构之间的对应关系提供可用于使用图形研究数据集的额外信息。在此基础上，我们对流形学习中的扩散映射进行了推广，并介绍了水平扩散映射的框架。我们将具有两两结构对应的数据集建模为配备连接的纤维丛。我们证明了加入这种附加信息的优势，并研究了一般纤维丛上的HDM的渐近行为。在更广泛的背景下，HDM揭示了高维数据集的次黎曼结构，并为具有结构对应的数据集提供了一个非参数学习框架。
Kernel-based nonlinear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph associated with the data set. When each individual data object carries additional structural details, however, the correspondence relations between these structures provide extra information that can be leveraged for studying the data set using the graph. Based on this observation, we generalizeDiffusion Maps(DM) in manifold learning and introduce the framework ofHorizontal Diffusion Maps(HDM). We model a data set with pairwise structural correspondences as afibre bundleequipped with aconnection. We demonstrate the advantage of incorporating such additional information and study the asymptotic behavior of HDM on general fibre bundles. In a broader context, HDM reveals the sub-Riemannian structure of high-dimensional data sets, and provides a nonparametric learning framework for data sets with structural correspondences.