Flip-flop spectrum-revealing QR factorization and its applications to singular value decomposition

Abstract

We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a significantly faster and more reliable variant of the QLP factorization of Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR factorization to initialize a partial column pivoted QR factorization and then compute a partial LQ factorization. As observed by Stewart in his original QLP work, Flip-Flop SRQR tracks the exact singular values with "considerable fidelity". We develop singular value lower bounds and residual error upper bounds for Flip-Flop SRQR factorization. In situations where singular values of the input matrix decay relatively quickly, the low-rank approximation computed by SRQR is guaranteed to be as accurate as truncated SVD. We also perform a complexity analysis to show that for the same accuracy, Flip-Flop SRQR is faster than randomized subspace iteration for approximating the SVD, the standard method used in Matlab tensor toolbox. We also compare Flip-Flop SRQR with alternatives on two applications, tensor approximation and nuclear norm minimization, to demonstrate its efficiency and effectiveness.