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
540-552
Markovsky, Ivan
7d632d37-2100-41be-a4ff-90b92752212c
Van Huffel, Sanine
9e98d33d-abae-491a-ab1c-eacc696d6f66
April 2007
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), .
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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wtls_note_proof.pdf
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wtls_note_answer.pdf
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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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