Dimension reduction algorithms for near-optimal low-dimensional embeddings and compressive sensing
Name
868329734-MIT.pdf
Description
Full printable version
Size
1.57 MB
Format
Adobe PDF
Checksum (MD5)
0bbdfbcd6b4d8a2ddfd06a05ecb1445b
Author(s)
Grant, Elyot
Advisor(s)
Piotr Indyk.
Date Issued
2013
Publisher
Massachusetts Institute of Technology
Abstract
In this thesis, we establish theoretical guarantees for several dimension reduction algorithms developed for applications in compressive sensing and signal processing. In each instance, the input is a point or set of points in d-dimensional Euclidean space, and the goal is to find a linear function from Rd into Rk , where k << d, such that the resulting embedding of the input pointset into k-dimensional Euclidean space has various desirable properties. We focus on two classes of theoretical results: -- First, we examine linear embeddings of arbitrary pointsets with the aim of minimizing distortion. We present an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ... is the smallest possible distortion over all orthonormal embeddings into k dimensions. This PTAS-like result transcends lower bounds for well-known embedding teclhniques such as the Johnson-Lindenstrauss transform. -- Next, motivated by compressive sensing of images, we examine linear embeddings of datasets containing points that are sparse in the pixel basis, with the goal of recoving a nearly-optimal sparse approximation to the original data. We present several algorithms that achieve strong recovery guarantees using the near-optimal bound of measurements, while also being highly "local" so that they can be implemented more easily in physical devices. We also present some impossibility results concerning the existence of such embeddings with stronger locality properties.
Description
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 41-42).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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