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.

使用子里曼几何学通过将点云的局部线性逼近组合成较低维度束，来进行流形学习和表面重构；通过获取由局部主成分分析得到的局部逼近构建在数学类R^d空间中低于维度d的级别k切子空间，我们称之为主切子空间，从而决定了R^d上的子里曼度量；我们展示了使用该度量进行子里曼测地线在一些重要问题上的应用，如：构建一个逼近子流形M，在R^k中构建点云的表示，以及在考虑到学习的几何特性时计算观测值之间的距离；当切空间精确估计时，重构保证与真实子流形相等，在模拟中展示了该框架在处理噪声数据时的稳健性；此框架还推广到基于先验已知里曼几何学的观测。

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