Randomized Rank-Revealing QLP for Low-Rank Matrix Decomposition

TL;DR

This paper introduces RU-QLP, a randomized, parallelizable low-rank matrix decomposition method that approximates SVD efficiently with theoretical guarantees and significant speedups on modern hardware.

Contribution

The paper proposes RU-QLP, a novel randomized partial QLP decomposition method that is highly parallelizable and provides theoretical bounds for low-rank approximation quality.

Findings

RU-QLP is rank-revealing and parallelizable.

Theoretical bounds are established for approximation accuracy.

RU-QLP achieves up to 7.1x speedup over randomized SVD.

Abstract

The pivoted QLP decomposition is computed through two consecutive pivoted QR decompositions, and provides an approximation to the singular value decomposition. This work is concerned with a partial QLP decomposition of low-rank matrices computed through randomization, termed Randomized Unpivoted QLP (RU-QLP). Like pivoted QLP, RU-QLP is rank-revealing and yet it utilizes random column sampling and the unpivoted QR decomposition. The latter modifications allow RU-QLP to be highly parallelizable on modern computational platforms. We provide an analysis for RU-QLP, deriving bounds in spectral and Frobenius norms on: i) the rank-revealing property; ii) principal angles between approximate subspaces and exact singular subspaces and vectors; and iii) low-rank approximation errors. Effectiveness of the bounds is illustrated through numerical tests. We further use a modern, multicore machine equipped with a GPU to demonstrate the efficiency of RU-QLP. Our results show that compared to the randomized SVD, RU-QLP achieves a speedup of up to 7.1 times on the CPU and up to 2.3 times with the GPU.