Riemannian流形光滑嵌入的切空间估计
Tangent space estimation for smooth embeddings of Riemannian manifolds
摘要：
数据分析中的许多降维问题涉及低维模型的恢复或数据集基础流形的学习。许多流形学习方法需要从局部可用的数据样本中估计流形在一点处的切空间。局部采样条件，如（i）邻域的大小（采样宽度）和（ii）邻域中的样本数量（采样密度）影响学习算法的性能。在这项工作中，我们提出了一个局部采样条件的理论分析，估计的切空间在一个点P躺在一个m维黎曼流形S在R^n。假设S在R^n中的光滑嵌入，我们通过对从S上P的邻域采样的点执行主成分分析（PCA）来估计切空间T_PS。我们的分析明确地考虑到在P流形的二阶属性，即主曲率以及高阶项。我们考虑一个随机抽样框架，并利用随机矩阵理论的最新成果，推导出采样宽度和局部采样密度的条件，以准确估计正切子空间。我们用估计的切空间与真切空间T_PS之间的夹角来度量估计精度，并给出了这个夹角以高概率有界的条件。特别是，我们观察到，局部采样条件是高度依赖于在二阶局部逼近的流形的组件之间的相关性。最后，我们提供了数值模拟来验证我们的理论研究结果。
英文摘要：
Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on a m-dimensional Riemannian manifold S in R^n. Assuming a smooth embedding of S in R^n, we estimate the tangent space T_P S by performing a Principal Component Analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second order properties of the manifold at P, namely the principal curvatures as well as the higher order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space and the true tangent space T_P S and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings.
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