Left vs right representations for solving weighted low-rank approximation problems
- 作者：Ivan Markovsky ; Sabine Van Huffel
- 关键词：Weighted low-rank approximation ; Total least squares ; Parameter optimization
- 刊名：Linear Algebra and its Applications
- 出版年：2007
- 期刊代码：38_00243795
- 类别：mt
- 出版时间：15 April 2007
- 卷：422
- 期：2-3
- 页码：540-552
- 文件大小：171 K
摘要
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.