AN IMPROVED ANALYSIS AND UNIFIED PERSPECTIVE ON DETERMINISTIC AND RANDOMIZED LOW-RANK MATRIX APPROXIMATION

We introduce a Generalized LU Factorization (GLU) for low-rank matrix approx-imation. 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 approx-imations. It also helps to explain the effect of sketching on the growth factor during Gaussian elimination.

Automated summary

We introduce a Generalized LU Factorization (GLU) for low-rank matrix approximation and compare it with past approaches. We provide complete bounds by combining established deterministic guarantees with sketching ensembles satisfying Johnson-Lindenstrauss properties. The subsampled randomized Hadamard transform (SRHT) ensemble shows particularly good performance. Moreover, the factorization unifies and generalizes many past algorithms, sometimes providing strictly better approximations, and helps explain the effect of sketching on the growth factor during Gaussian elimination.