Reconstruction and interpolation of manifolds I: The geometric Whitney problem

TL;DR

This paper addresses the problem of reconstructing Riemannian manifolds from metric spaces and point clouds, providing constructive solutions, characterizations, and algorithms with complexity estimates for manifold approximation in various geometric contexts.

Contribution

It offers new constructive methods and characterizations for approximating metric spaces by Riemannian manifolds with bounded geometry, and develops algorithms for manifold reconstruction with complexity analysis.

Findings

Characterization of metric spaces approximable by Riemannian manifolds with bounded geometry

New criteria for approximating Alexandrov spaces with curvature bounds

Algorithmic procedures with complexity estimates for manifold reconstruction

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 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 characterisation of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterise 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.