Local eigenvalue decomposition for embedded Riemannian manifolds
Local eigenvalue decomposition for embedded Riemannian manifolds
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DOI：
10.1016/j.laa.2020.06.006
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发表时间：
2020-11
影响因子：
1.1
通讯作者：
Javier Álvarez-Vizoso;M. Kirby;C. Peterson
中科院分区：
文献类型：
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作者：
Javier Álvarez-Vizoso;M. Kirby;C. Peterson
局部主成分分析可以在嵌入的黎曼流形的小区域上执行，以便将基础点集的协方差分析与局部外在和内在曲率相关联。我们表明，一般余维的子流形上的域的体积，由周围空间中的高维圆柱体和球的交叉，具有渐近展开的平均和标量曲率。此外，我们提出了一个推广的经典的第三基本形式一般子流形，并证明了局部特征值分解（EVD）的协方差矩阵的渐近展开，其中包含的曲率信息编码的这个张量的痕迹。这证明了局部EVD积分不变量与任意嵌入黎曼子流形的微分几何曲率之间的一般对应关系，到目前为止只发现曲线和超曲面。因此，我们建立了一个关键的理论桥梁，通过协方差矩阵的规模，在流形学习相关的统计数据的点云采样黎曼子流形的基础几何的潜在应用。
Local Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, have asymptotic expansions in terms of the mean and scalar curvatures. Moreover, we propose a generalization of the classical third fundamental form to general submanifolds and prove that the local eigenvalue decomposition (EVD) of the covariance matrices have asymptotic expansions that contain the curvature information encoded by the traces of this tensor. This proves the general correspondence between the local EVD integral invariants and differential-geometric curvature for arbitrary embedded Riemannian submanifolds, found so far for curves and hypersurfaces only. Thus, we establish a key theoretical bridge, via covariance matrices at scale, for potential applications in manifold learning relating the statistics of point clouds sampled from Riemannian submanifolds to the underlying geometry.