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Dissertation/Thesis Abstract

Sparse recovery and parameterization of manifold-valued data
by Taylor, Kye, M.S., University of Colorado at Boulder, 2008, 105; 1453576
Abstract (Summary)

Researchers are often confronted with high-dimensional datasets that contain some low-dimensional structure they would like to detect and emphasize with a more efficient description. Adding to the difficulties in recovering this structure in high dimensions is noise or uncertainty in the measurements that will be present in all real datasets. Despite this reality, there has been little exploration into how noise affects recovering the low-dimensional structure and possible ways to remedy these effects.

Several popular methods that can detect noise-free manifold signals in high-dimensions rely on approximations to the Laplace-Beltrami operator defined on the dataset. This operator encodes geometrical properties of the low-dimensional space where the data lies. Consequently, the eigenfunctions of the operator also contain information about the geometry of the space where the data originated. The set of eigenfunctions form a basis for square integrable functions defined on the dataset and converge to the eigenfunctions of the manifold's Laplacian as the distance between data points sampled from the manifold decreases.

In this thesis we explore recovering the eigenfunctions of the manifold's Laplacian when the data is corrupted by noise. Then, using the regularized eigenfunctions, we recover the true signal itself. In addition, we also examine parameterizing a dataset by relationships between local neighborhoods instead of by relationships between single data points alone. This emphasizes the local nature of the manifold structure, suppresses the impact of the noise on the parameterization, as well as opens up doors to multi-scale processing. We develop and analyze these ideas along with an iterative technique that simultaneously regularizes and parameterizes the manifold signal. This algorithm is tested on three different manifold signals: curves, surfaces, and images. For each data type, the proposed algorithm outperforms state-of-the-art methods for removing noise.

Indexing (document details)
Advisor: Meyer, Francois G.
Commitee: Curry, James H., Dougherty, Anne M.
School: University of Colorado at Boulder
Department: Applied Mathematics
School Location: United States -- Colorado
Source: MAI 46/06M, Masters Abstracts International
Subjects: Mathematics
Keywords: Denoising, Embedding, Laplacian eigenmaps, Manifold learning, Patch space
Publication Number: 1453576
ISBN: 978-0-549-56270-2
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