Hybrid iterative refined restarted Lanczos bidiagonalization methods

Explicit deflation in Golub-Kahan-Lanczos bidiagonalization methods

“…Matrices arising in applications such as principal component analysis (PCA) [18], genomics [1,32], data mining, data visualization, machine learning, pattern recognition [13], and directed networks [4], are often very large, sparse, and only accessible via matrix-vector routines, thus making it impractical for the computation of all singular triplets. Fortunately, with such matrices, one is often interested in computing only a few of the largest singular triplets-this has spurred a considerable amount of research (see, e.g., [6,7,19,20,21,24,25,31,33] and the references therein). This paper also deals with the computation of the largest singular triplets, though our starting point here is different when compared to the previously listed references.…”

“…The process of explicit deflation with subspace restriction for the GKLB process, although quite natural, does not appear to be well-investigated in the literature. We do remark that svdifp [24] and rd2svds [6] compute only one singular triplet at a time and apply an internal explicit deflation techniques to compute additional singular triplets. These differ from the focus of this paper.…”

“…These differ from the focus of this paper. For discussions on techniques for internal deflation (locking/purging) while computing the k 1 singular triplets, see for example [6,9,19,24].…”

“…Once an m-GKLB factorization is available, a variety of projected approximations can be used to estimate the singular triplets of A, e.g., Ritz methods [6,7,19,20,21,31], harmonic Ritz methods [7], (iterative) refined Ritz and refined harmonic Ritz methods [6,17,19]. In this paper, in order to keep our analysis short, we focus on the Ritz method as one of the most commonly used approximation for the largest singular triplets.…”