Manifold Curvature From Covariance Analysis
Manifold Curvature From Covariance Analysis
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DOI：
10.1109/ssp.2018.8450855
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发表时间：
2018-06
期刊：
影响因子：
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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
提出圆柱邻域的主成分分析来研究嵌入黎曼流形的局部几何。在每个通用点和尺度上，与该点处的切空间正交的高维圆柱体会切出一个路径连接的补丁，该补丁在环境空间中的点集分布编码了内在曲率和外在曲率。该邻域点的协方差矩阵具有特征向量，其尺度限制趋向于曲线的 Frenet-Serret 框架，以及我们所说的子流形的 Ricci-Weingarten 主方向。更重要的是，可以利用特征值的差和乘积的极限来恢复该点的曲率信息。用主曲率表示的超曲面公式特别简单，在高维情况的研究中起着至关重要的作用。
Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a high-dimensional cylinder orthogonal to the tangent space at the point cuts out a path-connected patch whose point-set distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the Frenet-Serret frame for curves, and to what we call the Ricci-Weingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases.