Sparse solution of underdetermined linear equations via adaptively iterative thresholding

Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is main...

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Vydáno v:	Signal processing Ročník 97; s. 152 - 161
Hlavní autoři:	Zeng, Jinshan, Lin, Shaobo, Xu, Zongben
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.04.2014 Elsevier
Témata:	Algorithms Applied sciences Coherence Convergence Detection, estimation, filtering, equalization, prediction Deviation Exact sciences and technology Global convergence Information, signal and communications theory Iterative methods Iterative thresholding algorithm Linear equations Mathematical analysis Mathematical models Signal and communications theory Signal, noise Sparse solution Telecommunications and information theory Underdetermined linear equations Underdetermined linear equations Iterative thresholding algorithm Global convergence Sparse solution Linear equation Threshold detection Coherence Signal processing Iterative method Algorithm Computational complexity Adaptive method
ISSN:	0165-1684, 1872-7557
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Abstract	Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is mainly due to their low computational complexities, especially for large scale applications. The aim of this paper is to provide guarantees on the global convergence of a wide class of iterative thresholding algorithms. Since the thresholds of the considered algorithms are set adaptively at each iteration, we call them adaptively iterative thresholding (AIT) algorithms. As the main result, we show that as long as A satisfies a certain coherence property, AIT algorithms can find the correct support set within finite iterations, and then converge to the original sparse solution exponentially fast once the correct support set has been identified. Meanwhile, we also demonstrate that AIT algorithms are robust to the algorithmic parameters. In addition, it should be pointed out that most of the existing iterative thresholding algorithms such as hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are included in the class of AIT algorithms studied in this paper. •We provide the convergence analysis of a class of adaptively iterative thresholding (AIT) algorithms for sparse solution of underdetermined linear equations y=Ax.•AIT algorithm converges to the unique sparsest solution with a linearly asymptotic convergence rate under the assumption that A satisfies a certain coherence property.•AIT algorithm finds the correct support set within finite iterations.•Most of the commonly used iterative thresholding algorithms are included in the class of AIT algorithms studied in this paper.
AbstractList	Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is mainly due to their low computational complexities, especially for large scale applications. The aim of this paper is to provide guarantees on the global convergence of a wide class of iterative thresholding algorithms. Since the thresholds of the considered algorithms are set adaptively at each iteration, we call them adaptively iterative thresholding (AIT) algorithms. As the main result, we show that as long as A satisfies a certain coherence property, AIT algorithms can find the correct support set within finite iterations, and then converge to the original sparse solution exponentially fast once the correct support set has been identified. Meanwhile, we also demonstrate that AIT algorithms are robust to the algorithmic parameters. In addition, it should be pointed out that most of the existing iterative thresholding algorithms such as hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are included in the class of AIT algorithms studied in this paper. Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is mainly due to their low computational complexities, especially for large scale applications. The aim of this paper is to provide guarantees on the global convergence of a wide class of iterative thresholding algorithms. Since the thresholds of the considered algorithms are set adaptively at each iteration, we call them adaptively iterative thresholding (AIT) algorithms. As the main result, we show that as long as A satisfies a certain coherence property, AIT algorithms can find the correct support set within finite iterations, and then converge to the original sparse solution exponentially fast once the correct support set has been identified. Meanwhile, we also demonstrate that AIT algorithms are robust to the algorithmic parameters. In addition, it should be pointed out that most of the existing iterative thresholding algorithms such as hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are included in the class of AIT algorithms studied in this paper. •We provide the convergence analysis of a class of adaptively iterative thresholding (AIT) algorithms for sparse solution of underdetermined linear equations y=Ax.•AIT algorithm converges to the unique sparsest solution with a linearly asymptotic convergence rate under the assumption that A satisfies a certain coherence property.•AIT algorithm finds the correct support set within finite iterations.•Most of the commonly used iterative thresholding algorithms are included in the class of AIT algorithms studied in this paper.
Author	Lin, Shaobo Xu, Zongben Zeng, Jinshan
Author_xml	– sequence: 1 givenname: Jinshan surname: Zeng fullname: Zeng, Jinshan email: jsh.zeng@gmail.com organization: School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China – sequence: 2 givenname: Shaobo surname: Lin fullname: Lin, Shaobo email: sblin1983@gmail.com organization: School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China – sequence: 3 givenname: Zongben surname: Xu fullname: Xu, Zongben email: zbxu@mail.xjtu.edu.cn organization: School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
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Keywords	Underdetermined linear equations Iterative thresholding algorithm Global convergence Sparse solution Linear equation Threshold detection Coherence Signal processing Iterative method Algorithm Computational complexity Adaptive method
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Snippet	Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number...
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SubjectTerms	Algorithms Applied sciences Coherence Convergence Detection, estimation, filtering, equalization, prediction Deviation Exact sciences and technology Global convergence Information, signal and communications theory Iterative methods Iterative thresholding algorithm Linear equations Mathematical analysis Mathematical models Signal and communications theory Signal, noise Sparse solution Telecommunications and information theory Underdetermined linear equations
Title	Sparse solution of underdetermined linear equations via adaptively iterative thresholding
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