## Low Rank Matrix Approximation Using The Lanczos Bidiagonalization Process With Applications (2000)

Venue: | SIAM J. Sci. Comput |

Citations: | 23 - 1 self |

### BibTeX

@ARTICLE{Simon00lowrank,

author = {Horst D. Simon and Hongyuan Zha},

title = {Low Rank Matrix Approximation Using The Lanczos Bidiagonalization Process With Applications},

journal = {SIAM J. Sci. Comput},

year = {2000},

volume = {21},

pages = {2257--2274}

}

### Years of Citing Articles

### OpenURL

### Abstract

Low rank approximation of large and/or sparse matrices is important in many applications. We show that good low rank matrix approximations can be directly obtained from the Lanczos bidiagonalization process without computing singular value decomposition. We also demonstrate that a so-called one-sided reorthogonalization process can be used to maintain adequate level of orthogonality among the Lanczos vectors and produce accurate low rank approximations. This technique reduces the computational cost of the Lanczos bidiagonalization process. We illustrate the efficiency and applicability of our algorithm using numerical examples from several applications areas.

### Citations

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Citation Context ...e considerable complication in forming approximations of Aj discussed in the previous section. We opt to use the approach that will maintain a certain level of orthogonality among the Lanczos vectors =-=[8, 16, 18, 19, 20]-=-. Even within this approach there exist several variations depending on how reorthogonalization is implemented. For example in SVDPACK, a state-of-the-art software package for computing dominant singu... |

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Citation Context ... of Aj. Lanczos bidiagonalization process has been used for computing a few dominant singular triplets (singular values and the corresponding left and right singular vectors) of large sparse matrices =-=[1, 3, 6, 8]-=-. We will show that in many cases of interest, good low-rank approximations of A can be directly obtained from the Lanczos bidiagonalization process of A without computing any SVD. We will also explor... |

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Citation Context ...or ill-posed problems, and latent semantic indexing in information retrieval for large document collections, to name a few, it is necessary to find a low-rank approximation of a given matrix A ∈R m×n =-=[4, 9, 17]-=-. Often A is a sparse or structured rectangular matrix, and sometimes either m ≫ n or m ≪ n. The theory of SVD provides the following characterization of the best rank-j approximation of A in terms of... |

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