Principal Sub-manifolds [article]

Zhigang Yao, Benjamin Eltzner, Tung Pham
2021 arXiv   pre-print
We invent a novel method of finding principal components in multivariate data sets that lie on an embedded nonlinear Riemannian manifold within a higher-dimensional space. Our aim is to extend the geometric interpretation of PCA, while being able to capture non-geodesic modes of variation in the data. We introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point on the manifold extending in the direction of highest variation in the
more » ... e spanned by the eigenvectors of the local tangent space PCA. Compared to recent work for the case where the sub-manifold is of dimension one –essentially a curve lying on the manifold attempting to capture one-dimensional variation–the current setting is much more general. The principal sub-manifold is therefore an extension of the principal flow, accommodating to capture higher dimensional variation in the data. We show the principal sub-manifold yields the ball spanned by the usual principal components in Euclidean space. By means of examples, we illustrate how to find, use and interpret a principal sub-manifold and we present an application in shape analysis.
arXiv:1604.04318v3 fatcat:a6xve7svbvfbvpb3hmksiuh5d4