Randomized QLP algorithm and error analysis

Abstract

In this paper, we describe the randomized QLP (RQLP) algorithm and its enhanced version (ERQLP) for computing the low rank approximation to $A$ of size $m\times n$ efficiently such that $A\approx QLP$, where $L$ is the rank-$k$ lower-triangular matrix, $Q$ and $P$ are column orthogonal matrices. The theoretical cost of the implementation of RQLP and ERQLP only needs $\mathcal{O}(mnk)$. Moreover, we derive the upper bounds of the expected approximation error $\mathbb{E}\left[({\sigma }_j(A)-{\sigma }_j(L))/{\sigma }_j(A)\right]$ for $j=1,\cdots ,k$, and prove that the $L$-values of the proposed methods can track the singular values of $A$ accurately. These claims are supported by extensive numerical experiments.