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Geometric manifold approximation using locally linear approximations
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Ahmed, Talal. Geometric manifold approximation using locally linear approximations. Retrieved from https://doi.org/doi:10.7282/T34J0H8N

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TitleGeometric manifold approximation using locally linear approximations
NameAhmed, Talal (author); Bajwa, Waheed U (chair); Patel, Vishal (internal member); Meer, Peter (internal member); Rutgers University; Graduate School - New Brunswick
Date Created2016
Other Date2016-05 (degree)
SubjectElectrical and Computer Engineering, Manifold (Learning theory)
Extent1 online resource (ix, 36 p. : ill.)
DescriptionThe 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.
NoteM.S.
NoteIncludes bibliographical references
Noteby Talal Ahmed
Genretheses, ETD graduate
Persistent URLhttps://doi.org/doi:10.7282/T34J0H8N
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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