Least upper bound of truncation error of low-rank matrix approximation algorithm using QR decomposition with pivoting

Abstract

Low-rank approximation by QR decomposition with pivoting (pivoted QR) is known to be less accurate than singular value decomposition (SVD); however, the calculation amount is smaller than that of SVD. The least upper bound of the ratio of the truncation error, defined by Vert A-BCVert _2, using pivoted QR to that using SVD is proved to be sqrt{frac{4^k-1}{3}(n-k)+1} for Ain {mathbb {R}}^{mtimes n}(mge n), approximated as a product of Bin {mathbb {R}}^{mtimes k} and Cin {mathbb {R}}^{ktimes n} in this study.