TY - JOUR

T1 - Reconstruction and interpolation of manifolds I: The geometric Whitney problem

AU - Fefferman, Charles

AU - Ivanov, Sergei

AU - Kurylev, Yaroslav

AU - Lassas, Matti

AU - Narayanan, Hariharan

N1 - Publisher Copyright: © 2019, The Author(s).

PY - 2020/10

Y1 - 2020/10

N2 - 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\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.

AB - 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\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.

KW - Tata Institute for Fundamental Research

KW - Riemannian manifolds

KW - Machine learning

KW - inverse problems

KW - Inverse problems

KW - Whitney’s extension problem

KW - M-SMOOTH FUNCTION

KW - Whitney's extension problem

KW - SET

KW - SPACES

KW - THEOREM

KW - EXTENSION PROBLEM

KW - NONLINEAR DIMENSIONALITY REDUCTION

KW - CONVERGENCE

KW - BOUNDARY

KW - EQUATION

KW - RIEMANNIAN-MANIFOLDS

UR - http://www.scopus.com/inward/record.url?scp=85075219918&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/reconstruction-interpolation-manifolds-i-geometric-whitney-problem

U2 - 10.1007/s10208-019-09439-7

DO - 10.1007/s10208-019-09439-7

M3 - Article

VL - 20

SP - 1035

EP - 1133

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 5

ER -