author: Simon, Horst D.; Zha, Hongyuan:
shorttitle: Low-rank matrix approximation using the Lanczos bidiagonalization
source: SIAM J. Sci. Comput. 21, No. 6, 2257 - 2274 (2000)
rsclass: 15A12; 65F10; 62P99; 62D05; 62-07; 62B05
keywords: low-rank approximation of matrices,
revtext: Approximations B (of rank j) of a given large non-square matrix A are studied. Their optimal form is defined as a minimizer of the Frobenius norm of the difference A - B. In principle, it can be constructed via a j-dimensional part (projection) of a certain diagonalized form of A. In such a setting one would need the full singular value decomposition of A (which generalizes the current diagonalization using two orthogonal matrices P and Q). The paper offers a cheaper algorithm based on the Lanczos bidiagonalization. The authors perform the error and stopping analysis and demonstrate that a mere one-sided re-orthogonalization of this process guarantees a sufficient precision. A collection of matrices from several application areas is used in illustrative tests.