Vandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-Resolution

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD)...

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Veröffentlicht in:	IEEE transactions on information theory Jg. 62; H. 6; S. 3685 - 3701
Hauptverfasser:	Yang, Zai, Xie, Lihua, Stoica, Petre
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York IEEE 01.06.2016 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	Algorithms Atomic measurements atomic norm Blocking Computer simulation Covariance matrices Decomposition Electronic mail Estimating techniques Frequency distribution Frequency estimation Mathematical models Matrix Matrix decomposition multidimensional frequency estimation Multilevel multilevel Toeplitz matrix Norms Numerical simulation Signal resolution Subspace methods superresolution The Vandermonde decomposition multilevel Toeplitz matrix The Vandermonde decomposition super-resolution multidimensional frequency estimation atomic norm
ISSN:	0018-9448, 1557-9654, 1557-9654
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Abstract	The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic \ell _{0} norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.
AbstractList	The Vandermonde decomposition of Toeplitz matrices, discovered by Caratheodory and Fejer in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic $\ell _{0}$ norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods. The Vandermonde decomposition of Toeplitz matrices, discovered by Caratheodory and Fejer in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic ... norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods. (ProQuest: ... denotes formulae/symbols omitted.) The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic \ell _{0} norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.
Author	Yang, Zai Stoica, Petre Xie, Lihua
Author_xml	– sequence: 1 givenname: Zai surname: Yang fullname: Yang, Zai email: yangzai@ntu.edu.sg organization: School of Automation, Nanjing University of Science and Technology, Nanjing, China – sequence: 2 givenname: Lihua surname: Xie fullname: Xie, Lihua email: elhxie@ntu.edu.sg organization: School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore – sequence: 3 givenname: Petre surname: Stoica fullname: Stoica, Petre email: ps@it.uu.se organization: Department of Information Technology, Uppsala University, Uppsala, Sweden
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Snippet	The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the... The Vandermonde decomposition of Toeplitz matrices, discovered by Caratheodory and Fejer in the 1910s and rediscovered by Pisarenko in the 1970s, forms the...
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SubjectTerms	Algorithms Atomic measurements atomic norm Blocking Computer simulation Covariance matrices Decomposition Electronic mail Estimating techniques Frequency distribution Frequency estimation Mathematical models Matrix Matrix decomposition multidimensional frequency estimation Multilevel multilevel Toeplitz matrix Norms Numerical simulation Signal resolution Subspace methods superresolution The Vandermonde decomposition
Title	Vandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-Resolution
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