Orientability and Diffusion Maps.
Orientability and Diffusion Maps.
复制标题
可定向性和扩散图。
DOI：
10.1016/j.acha.2010.10.001
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发表时间：
2011
影响因子：
2.5
通讯作者：
Wu,Hau-Tieng
中科院分区：
文献类型：
--
作者：
Singer,Amit;Wu,Hau-Tieng
分析高维大数据集的主要目的之一是了解其几何和拓扑结构。尽管数据本身被参数化为高维环境空间Rp中的点云，但参数之间的相关性往往表明了“流形假设”，即数据点分布在（或附近）嵌入Rp中的低维黎曼流形m上，且d≪p。在给定足够数量的采样数据点的情况下，我们介绍了一种确定本征流形可定向性的算法。如果流形是可定向的，那么我们的算法还提供了计算拉普拉斯特征函数的另一种方法，拉普拉斯特征函数在降低数据维数的扩散图框架中很重要。如果流形是不可定向的，那么我们给出了它的可定向双覆盖的一个改进的扩散映射。
One of the main objectives in the analysis of a high dimensional large data set is to learn its geometric and topological structure. Even though the data itself is parameterized as a point cloud in a high dimensional ambient space Rp, the correlation between parameters often suggests the “manifold assumption” that the data points are distributed on (or near) a low dimensional Riemannian manifold Mdembedded in Rp, with d≪p. We introduce an algorithm that determines the orientability of the intrinsic manifold given a sufficiently large number of sampled data points. If the manifold is orientable, then our algorithm also provides an alternative procedure for computing the eigenfunctions of the Laplacian that are important in the diffusion map framework for reducing the dimensionality of the data. If the manifold is non-orientable, then we provide a modified diffusion mapping of its orientable double covering.