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Covariance integral invariants of embedded Riemannian manifolds for manifold learning

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dc.contributor.authorÁlvarez Vizoso, Javier, author
dc.contributor.authorPeterson, Christopher, advisor
dc.contributor.authorKirby, Michael, advisor
dc.contributor.authorBates, Dan, committee member
dc.contributor.authorCavalieri, Renzo, committee member
dc.contributor.authorEykholt, Richard, committee member
dc.date.accessioned2018-09-10T20:04:22Z
dc.date.available2018-09-10T20:04:22Z
dc.date.issued2018
dc.description.abstractThis thesis develops an effective theoretical foundation for the integral invariant approach to study submanifold geometry via the statistics of the underlying point-set, i.e., Manifold Learning from covariance analysis. We perform Principal Component Analysis over a domain determined by the intersection of an embedded Riemannian manifold with spheres or cylinders of varying scale in ambient space, in order to generalize to arbitrary dimension the relationship between curvature and the eigenvalue decomposition of covariance matrices. In the case of regular curves in general dimension, the covariance eigenvectors converge to the Frenet-Serret frame and the corresponding eigenvalues have ratios that asymptotically determine the generalized curvatures completely, up to a constant that we determine by proving a recursion relation for a certain sequence of Hankel determinants. For hypersurfaces, the eigenvalue decomposition has series expansion given in terms of the dimension and the principal curvatures, where the eigenvectors converge to the Darboux frame of principal and normal directions. In the most general case of embedded Riemannian manifolds, the eigenvalues and limit eigenvectors of the covariance matrices are found to have asymptotic behavior given in terms of the curvature information encoded by the third fundamental form of the manifold, a classical tensor that we generalize to arbitrary dimension, and which is related to the Weingarten map and Ricci operator. These results provide descriptors at scale for the principal curvatures and, in turn, for the second fundamental form and the Riemann curvature tensor of a submanifold, which can serve to perform multi-scale Geometry Processing and Manifold Learning, making use of the advantages of the integral invariant viewpoint when only a discrete sample of points is available.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierxC1lvarezVizoso_colostate_0053A_14884.pdf
dc.identifier.urihttps://hdl.handle.net/10217/191299
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectcurvature
dc.subjectPCA
dc.subjectcovariance analysis
dc.subjectRiemannian manifold
dc.subjectintegral invariants
dc.titleCovariance integral invariants of embedded Riemannian manifolds for manifold learning
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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