University of Oulu

Expansions and factorizations of matrices and their applications

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Author: Pingamage Don, Charmin1
Organizations: 1University of Oulu, Faculty of Information Technology and Electrical Engineering, Department of Computer Science and Engineering, Computer Science
Format: ebook
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 3.2 MB)
Pages: 58
Persistent link: http://urn.fi/URN:NBN:fi:oulu-202006262686
Language: English
Published: Oulu : C. Pingamage Don, 2020
Publish Date: 2020-06-29
Thesis type: Master's thesis
Tutor: Huhtanen, Marko
Reviewer: Seppänen, Tapio
Huhtanen, Marko
Description:

Abstract

Linear algebra is a foundation to decompositions and algorithms for extracting simple structures from complex data. In this thesis, we investigate and apply modern techniques from linear algebra to solve problems arising in signal processing and computer science. In particular, we focus on data that takes the shape of a matrix and we explore how to represent it as products of circulant and diagonal matrices. To this end, we study matrix decompositions, approximations, and structured matrix expansions whose elements are products of circulant and diagonal matrices. Computationally, we develop a matrix expansion with DCD matrices for approximating a given matrix. Remarkably, DCD matrices, i.e., a product of diagonal matrix, circulant matrix, and another diagonal matrix, give an natural extension to rank-one matrices. Inspired from the singular value decomposition, we introduce a notion of a matrix rank closely related to the expansion and compute the rank of some specific structured matrices. Specifically, Toeplitz matrix is a sum of two DCD matrices. Here, we present a greedy algorithmic framework to devise the expansion numerically. Finally, we show that the practical uses of the DCD expansion can be complemented by the proposed framework and perform two experiments with a view towards applications.

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Copyright information: © Charmin Pingamage Don, 2020. This publication is copyrighted. You may download, display and print it for your own personal use. Commercial use is prohibited.