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  Manifold curvature learning from hypersurface integral invariants

Alvarez Vizoso, J., Kirby, M., & Peterson, C. (2020). Manifold curvature learning from hypersurface integral invariants. Linear Algebra and its Applications, 602, 179-205. doi:10.1016/j.laa.2020.05.020.

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Item Permalink: http://hdl.handle.net/21.11116/0000-0006-9736-3 Version Permalink: http://hdl.handle.net/21.11116/0000-0006-9737-2
Genre: Journal Article

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 Creators:
Alvarez Vizoso, Javier1, Author              
Kirby, Michael, Author
Peterson, Chris, Author
Affiliations:
1Max Planck Research Group and ERC Consolidator Grant: Solar and Stellar Dynamos - SOLSTAR, Max Planck Institute for Solar System Research, Max Planck Society, ou_2265638              

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 Abstract: Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Published online
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: DOI: 10.1016/j.laa.2020.05.020
 Degree: -

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Title: Linear Algebra and its Applications
Source Genre: Journal
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Publ. Info: New York : North-Holland
Pages: - Volume / Issue: 602 Sequence Number: - Start / End Page: 179 - 205 Identifier: ISSN: 0024-3795
CoNE: https://pure.mpg.de/cone/journals/resource/954925421092