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dc.contributor.advisorPeterson, Chris
dc.contributor.advisorKirby, Michael
dc.contributor.authorMarks, Justin D.
dc.contributor.committeememberBates, Dan
dc.contributor.committeememberAnderson, Chuck
dc.date.accessioned2007-01-03T08:26:22Z
dc.date.available2007-01-03T08:26:22Z
dc.date.submitted2012
dc.description2012 Fall.
dc.descriptionIncludes bibliographical references.
dc.description.abstractThe geometrically elegant Stiefel and Grassmann manifolds have become organizational tools for data applications, such as illumination spaces for faces in digital photography. Modern data analysis involves increasingly large-scale data sets, both in terms of number of samples and number of features per sample. In circumstances such as when large-scale data has been mapped to a Stiefel or Grassmann manifold, the computation of mean representatives for clusters of points on these manifolds is a valuable tool. We derive three algorithms for determining mean representatives for a cluster of points on the Stiefel manifold and the Grassmann manifold. Two algorithms, the normal mean and the projection mean, follow the theme of the Karcher mean, relying upon inversely related maps that operate between the manifold and the tangent bundle. These maps are informed by the geometric definition of the tangent bundle and the normal bundle. From the cluster of points, each algorithm exploits these maps in a predictor/corrector loop until converging, with prescribed tolerance, to a fixed point. The fixed point acts as the normal mean representative, or projection mean representative, respectively, of the cluster. This method shares its principal structural characteristics with the Karcher mean, but utilizes a distinct pair of inversely related maps. The third algorithm, called the flag mean, operates in a context comparable to a generalized Grassmannian. It produces a mean subspace of arbitrary dimension. We provide applications and discuss generalizations of these means to other manifolds.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierMarks_colostate_0053A_11461.pdf
dc.identifierETDF2012500311MATH
dc.identifier.urihttp://hdl.handle.net/10217/71570
dc.languageEnglish
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019 - CSU Theses and Dissertations
dc.rightsCopyright of the original work is retained by the author.
dc.subjectGrassmann
dc.subjectmanifold
dc.subjectmean
dc.subjectStiefel
dc.subjectsubspace
dc.titleMean variants on matrix manifolds
dc.typeText
dcterms.rights.dplaThe copyright and related rights status of this Item has not been evaluated (https://rightsstatements.org/vocab/CNE/1.0/). Please refer to the organization that has made the Item available for more information.
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)


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