A New Look at the Power Method for Fast Subspace Tracking
Identifieur interne : 001A66 ( Istex/Corpus ); précédent : 001A65; suivant : 001A67A New Look at the Power Method for Fast Subspace Tracking
Auteurs : Yingbo Hua ; Yong Xiang ; Tianping Chen ; Karim Abed-Meraim ; Yongfeng MiaoSource :
- Digital Signal Processing [ 1051-2004 ] ; 1999.
English descriptors
- KwdEn :
- Algorithm, Arbitrary initializations, Asymmetric square root, Convergence, Convergent, Converges, Covariance, Covariance matrix, Eigenvalue, Eigenvector, Exponential convergence property, Ieee, Ieee trans, Initial matrix, Iteration, Leakage factor, Malase, Matrix, Mild condition, Natural power method, Neural networks, Novel information criterion, Orthogonality, Orthonormal, Other methods, Past method, Performance comparison, Power method, Previous estimate, Principal component, Principal components, Principal eigenvector, Principal subspace, Projection approximation, Projection approximation subspace, Random vectors, Right side, Second term, Signal process, Signal processing, Small step size, Step size, Subspace, Trans, Unitary, Unitary matrix, Unitary rotation, Vector sequence, Weight matrix, Weight vector.
- Teeft :
- Algorithm, Arbitrary initializations, Asymmetric square root, Convergence, Convergent, Converges, Covariance, Covariance matrix, Eigenvalue, Eigenvector, Exponential convergence property, Ieee, Ieee trans, Initial matrix, Iteration, Leakage factor, Malase, Matrix, Mild condition, Natural power method, Neural networks, Novel information criterion, Orthogonality, Orthonormal, Other methods, Past method, Performance comparison, Power method, Previous estimate, Principal component, Principal components, Principal eigenvector, Principal subspace, Projection approximation, Projection approximation subspace, Random vectors, Right side, Second term, Signal process, Signal processing, Small step size, Step size, Subspace, Trans, Unitary, Unitary matrix, Unitary rotation, Vector sequence, Weight matrix, Weight vector.
Abstract
Abstract: A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.
Url:
DOI: 10.1006/dspr.1999.0348
Links to Exploration step
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<front><div type="abstract" xml:lang="en">Abstract: A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.</div>
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<abstract>Abstract: A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.</abstract>
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<abstract xml:lang="en"><p>A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.</p>
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<head><ce:dochead><ce:textfn>Regular Article</ce:textfn>
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<ce:title>A New Look at the Power Method for Fast Subspace Tracking</ce:title>
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<ce:note-para>E-mail: yhua@ee.mu.oz.au</ce:note-para>
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<ce:author><ce:given-name>Yong</ce:given-name>
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<ce:author><ce:given-name>Tianping</ce:given-name>
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<ce:author><ce:given-name>Yongfeng</ce:given-name>
<ce:surname>Miao</ce:surname>
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<ce:textfn>Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Victoria, 3052, Australia, Fudan University, China, Paris, France, California , USA</ce:textfn>
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<ce:textfn>University of Melbourne, Parkville, Victoria, 3052, Australia, Department of Mathematics, Fudan University, China, Paris, France, California , USA</ce:textfn>
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<ce:abstract><ce:section-title>Abstract</ce:section-title>
<ce:abstract-sec><ce:simple-para>A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively <ce:italic>O</ce:italic>
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is the dimension of the vector sequence and <ce:italic>p</ce:italic>
is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The <ce:italic>O</ce:italic>
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<abstract lang="en">Abstract: A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.</abstract>
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