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.

翻译：本文展示了如何将次黎曼几何用于流形学习与曲面重建，通过结合点云的局部线性近似来构建低维丛。通过局部主成分分析获得的局部近似被整合为 $\mathbb{R}^d$（$k<d$）上的秩 $k$ 切子丛，我们称之为主丛。这定义了 $\mathbb{R}^d$ 上的一个次黎曼度量。我们证明，针对该度量的次黎曼测地线可成功应用于若干重要问题，包括：近似子流形 $M$ 的显式构造、$\mathbb{R}^k$ 中点云表示的构建，以及考虑所学几何结构后观测值间距离的计算。在切空间被精确估计的极限情况下，该重建保证等于真实子流形。通过模拟实验，我们表明该框架在处理噪声数据时具有鲁棒性。此外，该框架可推广至先验已知黎曼流形上的观测。