Low-rank approximation: Randomized QR with Column Pivoting and related methods using sparse projection and pass-efficient techniques

Currently, several efficient algorithms utilize randomization techniques to compute low-rank matrix approximations, such as Randomized QR with Column Pivoting (RQRCP) and Flip-Flop spectrum-revealing QR (FFSRQR). While these algorithms have achieved notable efficiency in dealing with large-scale low-rank matrix approximations, there is still scope for improvement. This paper introduces an improved version of the RQRCP algorithm, enhanced by incorporating a sparse embedding matrix (SEM) for sparse projection, referred to as RQRCP-SEM. Furthermore, building on RQRCP-SEM and employing pass-efficient techniques, this paper proposes two versions of the PEFFSRQR-SEM algorithm to further optimize the efficiency of the FFSRQR algorithm. The paper theoretically analyzes the approximation error and computational complexity of these new algorithms and validates these analyses through numerical experiments.