Mean variants on matrix manifolds
| dc.contributor.author | Marks, Justin D., author | |
| dc.contributor.author | Peterson, Chris, advisor | |
| dc.contributor.author | Kirby, Michael, advisor | |
| dc.contributor.author | Bates, Dan, committee member | |
| dc.contributor.author | Anderson, Chuck, committee member | |
| dc.date.accessioned | 2007-01-03T08:26:22Z | |
| dc.date.available | 2007-01-03T08:26:22Z | |
| dc.date.issued | 2012 | |
| dc.description.abstract | The 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.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Marks_colostate_0053A_11461.pdf | |
| dc.identifier | ETDF2012500311MATH | |
| dc.identifier.uri | http://hdl.handle.net/10217/71570 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright 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.subject | Grassmann | |
| dc.subject | manifold | |
| dc.subject | mean | |
| dc.subject | Stiefel | |
| dc.subject | subspace | |
| dc.title | Mean variants on matrix manifolds | |
| dc.type | Text | |
| dcterms.rights.dpla | This 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.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
Files
Original bundle
1 - 1 of 1
Loading...
- Name:
- Marks_colostate_0053A_11461.pdf
- Size:
- 1.33 MB
- Format:
- Adobe Portable Document Format
- Description: