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Reconstruction and interpolation of manifolds I: The geometric Whitney problem | | Charles Fefferman
; Sergei Ivanov
; Yaroslav Kurylev
; Matti Lassas
; Hariharan Narayanan
; | | Date: |
4 Aug 2015 | | 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 $Ssubset {mathbb R}^m$, $m>n$ 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. | | Source: | arXiv, 1508.0674 | | Services: | Forum | Review | PDF | Favorites | |
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