eprintid: 1522488
rev_number: 21
eprint_status: archive
userid: 608
dir: disk0/01/52/24/88
datestamp: 2017-10-18 12:40:44
lastmod: 2021-10-10 23:13:38
status_changed: 2017-10-18 12:40:44
type: working_paper
metadata_visibility: show
creators_name: Fefferman, C
creators_name: Ivanov, S
creators_name: Kurylev, Y
creators_name: Lassas, M
creators_name: Narayanan, H
title: Reconstruction and interpolation of manifolds I: The geometric Whitney problem
ispublished: unpub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
keywords: Whitney’s extension problem, Riemannian manifolds, machine learning, inverse problems
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 learning) where a smooth $n$-dimensional surface $S\subset {\mathbb R}^m$, $m>n$ 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. 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. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in ${\mathbb R}^m$ and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data.
date: 2015-08-04
official_url: http://arxiv.org/abs/1508.00674v1
oa_status: green
full_text_type: other
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 1044952
lyricists_name: Kurylev, Yaroslav
lyricists_id: YKURL27
actors_name: Kurylev, Yaroslav
actors_id: YKURL27
actors_role: owner
full_text_status: public
pages: 49
citation:        Fefferman, C;    Ivanov, S;    Kurylev, Y;    Lassas, M;    Narayanan, H;      (2015)    Reconstruction and interpolation of manifolds I: The geometric Whitney problem.                           Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/1522488/1/1508.00674v1.pdf