A new randomized CholeskyQR based on LU decomposition with partial pivoting

Abstract

CholeskyQR has received considerable attention in recent years for its efficiency and simplicity in computing QR decomposition of the tall-skinny $X \in \mathbb{R}^{m\times n}$ with $m \ge n$ and $\mbox{rank}(X)=n$. Leveraging matrix sketching from randomized linear algebra, randomized CholeskyQR (RCholeskyQR) has been proposed to accelerate the computation by reducing the dimension of the problems. In this work, we propose RCLUPP, a new randomized CholeskyQR-type algorithm based on LU decomposition with partial pivoting (LUP decomposition). By taking LUP decomposition and the thin HouseholderQR on the sketched matrix, RCLUPP significantly improves the applicability and efficiency compared with LU-CholeskyQR2 (LC2). We present a rigorous rounding error analysis of RCLUPP, with a sharper bound of residual compared with those in the existing works. Comparative studies demonstrate that RCLUPP outperforms CholeskyQR2, Shifted CholeskyQR3 (SCholeskyQR3), and LC2 in terms of applicability while maintaining competitive accuracy and efficiency. A variant, RCLUPPr, performs LUP decomposition directly on $X \in \mathbb{R}^{m\times n}$, offering exceptional robustness and numerical stability for the ill-conditioned scenarios, which exceeds that of RCLUPP and RCholeskyQR. Numerical experiments on the synthetic and real-world matrices validate the theoretical results.