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Left vs right representations for solving weighted low-rank approximation problems

Left vs right representations for solving weighted low-rank approximation problems
Left vs right representations for solving weighted low-rank approximation problems
The weighted low-rank approximation problem in general has no analytical solution in terms of the singular value decomposition and is solved numerically using optimization methods. Four representations of the rank constraint that turn the abstract problem formulation into parameter optimization problems are presented. The parameter optimization problem is partially solved analytically, which results in an equivalent quadratically constrained problem. A commonly used re-parameterization avoids the quadratic constraint and makes the equivalent problem a nonlinear least squares problem, however, it might be necessary to change this re-parameterization during the iteration process. It is shown how the cost function can be computed efficiently in two special cases: row-wise and column-wise weighting.
Weighted low-rank approximation, Total least squares, Parameter optimization
0024-3795
540-552
Markovsky, Ivan
7d632d37-2100-41be-a4ff-90b92752212c
Van Huffel, Sanine
9e98d33d-abae-491a-ab1c-eacc696d6f66
Markovsky, Ivan
7d632d37-2100-41be-a4ff-90b92752212c
Van Huffel, Sanine
9e98d33d-abae-491a-ab1c-eacc696d6f66

Markovsky, Ivan and Van Huffel, Sanine (2007) Left vs right representations for solving weighted low-rank approximation problems. Linear Algebra and Its Applications, 422 (2-3), 540-552.

Record type: Article

Abstract

The weighted low-rank approximation problem in general has no analytical solution in terms of the singular value decomposition and is solved numerically using optimization methods. Four representations of the rank constraint that turn the abstract problem formulation into parameter optimization problems are presented. The parameter optimization problem is partially solved analytically, which results in an equivalent quadratically constrained problem. A commonly used re-parameterization avoids the quadratic constraint and makes the equivalent problem a nonlinear least squares problem, however, it might be necessary to change this re-parameterization during the iteration process. It is shown how the cost function can be computed efficiently in two special cases: row-wise and column-wise weighting.

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More information

Published date: April 2007
Keywords: Weighted low-rank approximation, Total least squares, Parameter optimization
Organisations: Southampton Wireless Group

Identifiers

Local EPrints ID: 263421
URI: http://eprints.soton.ac.uk/id/eprint/263421
ISSN: 0024-3795
PURE UUID: bbec7146-bbbc-4ea7-b307-d2a489ac8d77

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Date deposited: 13 Feb 2007
Last modified: 14 Mar 2024 07:31

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Contributors

Author: Ivan Markovsky
Author: Sanine Van Huffel

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