Lanczos, Householder transformations, and implicit deflation for fast and reliable dominant singular subspace computation

Abstract

AbstractMany applications, such as subspace‐based models in information retrieval and signal processing, require the computation of singular subspaces associated with the k dominant, or largest, singular values of an m×n data matrix A, where k≪min(m,n). Frequently, A is sparse or structured, which usually means matrix–vector multiplications involving A and its transpose can be done with much less than 𝒪(mn) flops, and A and its transpose can be stored with much less than 𝒪(mn) storage locations. Many Lanczos‐based algorithms have been proposed through the years because the underlying Lanczos method only accesses A and its transpose through matrix–vector multiplications. We implement a new algorithm, called KSVD, in the Matlab environment for computing approximations to the singular subspaces associated with the k dominant singular values of a real or complex matrix A. KSVD is based upon the Lanczos tridiagonalization method, the WY representation for storing products of Householder transformations, implicit deflation, and the QR factorization. Our Matlab simulations suggest it is a fast and reliable strategy for handling troublesome singular‐value spectra. Copyright © 2001 John Wiley & Sons, Ltd.