The design and analysis of methods in signal processing is greatly impacted by the model being selected to represent the signals of interest. For many decades, the most popular geometric models in signal processing have been the subspace and the union-of-subspaces signal models. However, there are classes of signals that are not well-represented by either the subspace or the union-of-subspaces model, but are manifested in a low-dimensional non-linear manifold embedded in a high-dimensional ambient space. Though a lot of work has been done on low-dimensional embedding of manifold sampled data, few works address the problem of approximating the manifold geometry in the ambient space. There is value in capturing the geometric variations of a non-linear manifold in the ambient space, as shown in this work. In this work, the local linearity of a manifold is exploited to address the problem of approximating the geometric structure of a non-linear non-intersecting manifold using a union of tangent planes (affine subspaces). The number of approximating tangent planes adapts with the varying manifold curvature such that the manifold geometry is always well approximated within a given accuracy. Also, the local linearity of the manifold is exploited to subsample the manifold data before using it to learn the manifold geometry, with negligible loss of approximation accuracy. Owing to this subsampling feature, the proposed approach shows more than a $100$-times decrease in the learning time when compared to state-of-the-art manifold learning algorithms, while achieving similar approximation accuracy. Because the approximating tangent planes extend indefinitely in space, the data encoding problem becomes complicated in the aforementioned learning approach. Thus, in the second half of the thesis, the manifold approximation problem is reformulated such that the geometry is approximated using a union of tangent patches, instead of tangent planes. Then, the data encoding problem is formulated as a series of convex optimization problems, and an efficient solution is proposed to solve each of the convex problems. Last, the value of capturing manifold geometry is demonstrated by showing the denoising performance of our proposed framework on both synthetic and real data.
Subject (authority = RUETD)
Topic
Electrical and Computer Engineering
Subject (authority = ETD-LCSH)
Topic
Manifold (Learning theory)
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_7280
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (ix, 36 p. : ill.)
Note (type = degree)
M.S.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Talal Ahmed
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.