Low-Rank Matrix Approximation Using the Lanczos Bidiagonalization Process with Applications
Low-Rank Matrix Approximation Using the Lanczos Bidiagonalization Process with Applications
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DOI：
10.1137/s1064827597327309
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发表时间：
1999-12
期刊：
影响因子：
--
通讯作者：
H. Simon;H. Zha
中科院分区：
文献类型：
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作者：
H. Simon;H. Zha
大矩阵和稀疏矩阵的低秩逼近在许多应用中是重要的，奇异值分解（SVD）给出了关于酉不变范数的最佳低秩逼近。在本文中，我们表明，良好的低秩近似可以直接从Lanczos双对角化过程中应用到给定的矩阵，而无需计算任何SVD。我们还证明了所谓的单侧reorthogonalization过程可以用来保持足够水平的Lanczos向量之间的正交性，并产生准确的低秩近似。这种技术降低了Lanczos双对角化过程的计算成本。我们说明了我们的算法的效率和适用性，从几个应用领域的数值例子。
Low-rank approximation of large and/or sparse matrices is important in many applications, and the singular value decomposition (SVD) gives the best low-rank approximations with respect to unitarily-invariant norms. In this paper we show that good low-rank approximations can be directly obtained from the Lanczos bidiagonalization process applied to the given matrix without computing any SVD. We also demonstrate that a so-called one-sided reorthogonalization process can be used to maintain an 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.