Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem
- PDF / 1,545,145 Bytes
- 99 Pages / 439.37 x 666.142 pts Page_size
- 104 Downloads / 176 Views
Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem Charles Fefferman1 · Sergei Ivanov2 · Yaroslav Kurylev3 · Matti Lassas4 · Hariharan Narayanan5
© The Author(s) 2019
Abstract We study the geometric Whitney problem on how a Riemannian manifold (M, g) 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 ⊂ Rm , m > n needs to be constructed to approximate a point cloud in Rm . 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 Rm and interpolated to a smooth submanifold. Keywords Whitney’s extension problem · Riemannian manifolds · Machine learning · Inverse problems
Communicated by Shmuel Weinberger. Extended author information available on the last page of the article
123
Foundations of Computational Mathematics
Mathematics Subject Classification 53C21 · 62G08 · 35R30 · 41A05
1 Introduction and the Main Results 1.1 Geometrization of Whitney’s Extension Problem In this paper, we develop a geometric version of Whitney’s extension problem. Let f : K → R be a function defined on a given (arbitrary) set K ⊂ Rn , and let m ≥ 1 be a given integer. The classical Whitney problem is the question whether f extends to a function F ∈ C m (Rn ) and if such an F exists, what is the optimal C m norm of the extension. Furthermore, one is interested in the questions if the derivatives of F, up to order m, at a given point can be estimated, or if one can construct extensi
Data Loading...
