Principal subbundles for dimension reduction
This provides a novel geometric framework for dimension reduction and manifold learning, addressing challenges in data analysis for fields like computer vision and machine learning, though it appears incremental as it builds on existing PCA and sub-Riemannian methods.
The paper tackles the problem of manifold learning and surface reconstruction from point clouds by introducing principal subbundles derived from local PCA approximations, which define a sub-Riemannian metric for computing geodesics. It demonstrates applications in constructing approximating submanifolds, representing data in lower dimensions, and computing distances, with robustness shown via simulations on noisy data and generalization to known Riemannian manifolds.
In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank $k$ tangent subbundle on $\mathbb{R}^d$, $k<d$, which we call a principal subbundle. This determines a sub-Riemannian metric on $\mathbb{R}^d$. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold $M$, construction of a representation of the point-cloud in $\mathbb{R}^k$, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.