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Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2019-11-13 , DOI: 10.1007/s10208-019-09439-7
Charles Fefferman , Sergei Ivanov , Yaroslav Kurylev , Matti Lassas , Hariharan Narayanan

We study the geometric Whitney problem on how a Riemannian manifold (Mg) can be constructed to approximate a metric space \((X,d_X)\). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold \(S\subset {{\mathbb {R}}}^m\), \(m>n\) needs to be constructed to approximate a point cloud in \({{\mathbb {R}}}^m\). These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in \({{\mathbb {R}}}^m\) and interpolated to a smooth submanifold.



中文翻译:

流形的重构和插值。一:几何惠特尼问题

我们研究了几何惠特尼问题,关于如何构造黎曼流形(M,  g)来近似度量空间\((X,d_X)\)。此问题与流形插值(或流形重构)密切相关,其中需要构造光滑的n维子流形\(S \ subset {{\ mathbb {R}}} ^ m \)\(m> n \)逼近\({{\ mathbb {R}}} ^ m \)中的点云。这些问题在微分几何,机器学习以及应用程序中遇到的许多逆问题中都会遇到。黎曼流形的确定包括其拓扑结构,可微结构和度量。我们为上述问题提供了建设性的解决方案。此外,我们用有界几何的黎曼流形表征了可以近似的度量空间:我们提供了充分的条件,以确保在Gromov–Hausdorff或准等距意义上,度量空间可以由a的黎曼流形近似。固定尺寸,并具有一定的直径,截面曲率和内射半径。此外,我们显示出需要修改参数值的类似条件。作为主要结果的应用,我们给出了带有两侧曲率边界的亚历山大空间的新刻画。此外,我们表征了欧几里德空间的子集,该子集可以在Hausdorff度量中通过固定维数的子流形和有界的主曲率和法向注入半径来近似。我们开发了算法程序来解决度量空间的几何惠特尼问题和欧氏空间中的流形重构问题,并估计这些程序的计算复杂性。还对无界度量集和流形研究了上述插值问题。黎曼流形的结果基于惠特尼嵌入结构的一般化,其中将近似坐标图嵌入到其中 我们表征欧几里得空间的子集,该子集可以在Hausdorff度量中通过固定维数的子流形和有界的主曲率和法向注入半径来近似。我们开发了算法程序来解决度量空间的几何惠特尼问题和欧氏空间中的流形重构问题,并估计这些程序的计算复杂性。还对无界度量集和流形研究了上述插值问题。黎曼流形的结果基于惠特尼嵌入结构的一般化,其中将近似坐标图嵌入到其中 我们表征欧几里得空间的子集,该子集可以在Hausdorff度量中通过固定维数的子流形和有界的主曲率和法向注入半径来近似。我们开发了算法程序来解决度量空间的几何惠特尼问题和欧氏空间中的流形重构问题,并估计这些程序的计算复杂性。还对无界度量集和流形研究了上述插值问题。黎曼流形的结果基于惠特尼嵌入结构的一般化,其中将近似坐标图嵌入到其中 我们开发了算法程序来解决度量空间的几何惠特尼问题和欧氏空间中的流形重构问题,并估计这些程序的计算复杂性。还对无界度量集和流形研究了上述插值问题。黎曼流形的结果基于惠特尼嵌入结构的一般化,其中将近似坐标图嵌入到其中 我们开发了算法程序来解决度量空间的几何惠特尼问题和欧氏空间中的流形重构问题,并估计这些程序的计算复杂性。还对无界度量集和流形研究了上述插值问题。黎曼流形的结果基于惠特尼嵌入结构的一般化,其中将近似坐标图嵌入到其中\({{\ mathbb {R}}} ^ m \)并插值到平滑子流形。

更新日期:2019-11-13
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