A weighted-least-squares matrix decomposition method with application to the design of two-dimensional digital filters

Shpak, D.J.

doi:10.1109/mwscas.1990.140910

Cited by 14 publications

(20 citation statements)

References 3 publications

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“…Theorem 13 (Quadratic Approximation): Define as in (5) and, having fixed and , define as in (10). Then, the second-order Taylor series approximation of about is vec grad vec vec vec (26) where grad is defined in (22), and is the symmetric matrix in (27), shown at the bottom of the page.…”

Section: Second-order Descent Methodsmentioning

confidence: 99%

“…A disadvantage of using the SVD to decompose the desired frequency response is that it treats all entries of equally, which in some cases leads to degraded designs. In order to discriminate between the important and unimportant elements of , the idea of replacing the SVD with a weighted low-rank approximation was proposed in [16], [27]. (See [16] for a design example.…”

Section: A Applicationsmentioning

confidence: 99%

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The geometry of weighted low-rank approximations

Manton

Mahony

Hua

2003

IEEE Trans. Signal Process.

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Abstract-The low-rank approximation problem is to approximate optimally, with respect to some norm, a matrix by one of the same dimension but smaller rank. It is known that under the Frobenius norm, the best low-rank approximation can be found by using the singular value decomposition (SVD). Although this is no longer true under weighted norms in general, it is demonstrated here that the weighted low-rank approximation problem can be solved by finding the subspace that minimizes a particular cost function. A number of advantages of this parameterization over the traditional parameterization are elucidated. Finding the minimizing subspace is equivalent to minimizing a cost function on the Grassmann manifold. A general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework. Within this framework, two novel algorithms (a steepest descent algorithm and a Newton-like algorithm) are derived for solving the weighted low-rank approximation problem. They are compared with other algorithms for low-rank approximation as well as with other algorithms for minimizing a cost function on a Grassmann manifold.Index Terms-Grassman manifold, low-rank approximations, optimization on manifolds, reduced rank signal processing.

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Section: Second-order Descent Methodsmentioning

confidence: 99%

Section: A Applicationsmentioning

confidence: 99%

The geometry of weighted low-rank approximations

Manton

Mahony

Hua

2003

IEEE Trans. Signal Process.

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“…As such, the application of a general nonlinear optimization algorithm usually yields a local minimizer and the performance of the local minimizer depends largely on the choice of the optimization algorithm and the initial point [14] [15]. However, there is an exception: if W 0 is an allone matrix, then a global minimizer of function J(x, W 0 ) in (6) is provided by the SVD of F [2].…”

Section: Wlra Of a Complex-valued Matrixmentioning

confidence: 99%

“…The WLRA was considered in [14] for real-valued matrices and in [15] for complex-valued matrices. However, the methods proposed in [14][15] only produce suboptimal solutions.…”

Section: Introductionmentioning

confidence: 99%

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New method for weighted low-rank approximation of complex-valued matrices and its application for the design of 2-D digital filters

Antoniou

Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03.

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The design of two-dimensional (2-D) digital filters can be accomplished using the singular-value decomposition (SVD) method proposed by the authors in the past. The method in its present form treats all the elements of the sampled frequency-response matrix uniformly. Although the method works very well, in certain applications improved designs can be achieved by preconditioning the frequencyresponse matrix in order to emphasize important parts and deemphasize unimportant parts of the matrix. The preconditioning can be achieved through the use of an optimal weighted low-rank approximation (WLRA). Current methods for WLRA provide only local solutions. In this paper, we propose a method that can be used to perform WLRA which is globally optimal for complex-valued matrices. The usefulness of the proposed method will be demonstrated by applying it to the design of 2-D digital filters.

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Weighted low-rank approximation of general complex matrices and its application in the design of 2-D digital filters

Pei²,

Wang³

1997

IEEE Trans. Circuits Syst. I

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In this brief we present a method for the weighted low-rank approximation of general complex matrices along with an algorithmic development for its computation. The method developed can be viewed as an extension of the conventional singular value decomposition to include a nontrivial weighting matrix in the approximation error measure. It is shown that the optimal rank-K weighted approximation can be achieved by computing K generalized Schmidt pairs and an iterative algorithm is presented to compute them. Application of the proposed algorithm to the design of FIR two-dimensional (2-D) digital filters is described to demonstrate the usefulness of the algorithm proposed. Index Terms-2-D digital filters, singular value decomposition.

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A weighted-least-squares matrix decomposition method with application to the design of two-dimensional digital filters

Cited by 14 publications

References 3 publications

The geometry of weighted low-rank approximations

The geometry of weighted low-rank approximations

New method for weighted low-rank approximation of complex-valued matrices and its application for the design of 2-D digital filters

Weighted low-rank approximation of general complex matrices and its application in the design of 2-D digital filters

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