An Improved Analysis and Unified Perspective on Deterministic and Randomized Low-Rank Matrix Approximation

 James DemmelLaura Grigori

We introduce a Generalized LU Factorization (GLU) for low-rank matrix approximation. We relate this to past approaches and extensively analyze its approximation properties. The established deterministic guarantees are combined with sketching ensembles satisfying Johnson–Lindenstrauss properties to present complete bounds. Particularly good performance is shown for the subsampled randomized Hadamard transform (SRHT) ensemble. Moreover, the factorization is shown to unify and generalize many past algorithms, sometimes providing strictly better approximations. It also helps to explain the effect of sketching on the growth factor during Gaussian elimination.