Parallel QR Factorization of Block Low-Rank Matrices

TL;DR

This paper introduces two novel parallel algorithms for QR factorization of Block Low-Rank matrices, demonstrating significant speedups and robustness over traditional dense QR methods on large matrices.

Contribution

The paper presents two new parallel Householder QR algorithms tailored for BLR matrices, improving efficiency and robustness compared to existing methods.

Findings

BLR-QR methods are over ten times faster than dense QR in MKL.

Parallel tiled BLR-QR achieves 50x speedup on 64-core CPU.

The proposed methods handle ill-conditioned matrices better than MGS-based approaches.

Abstract

We present two new algorithms for Householder QR factorization of Block Low-Rank (BLR) matrices: one that performs block-column-wise QR, and another that is based on tiled QR. We show how the block-column-wise algorithm exploits BLR structure to achieve arithmetic complexity of $\mathcal{O}(mn)$, while the tiled BLR-QR exhibits $\mathcal{O}(mn^{1.5})$ complexity. However, the tiled BLR-QR has finer task granularity that allows parallel task-based execution on shared memory systems. We compare the block-column-wise BLR-QR using fork-join parallelism with tiled BLR-QR using task-based parallelism. We also compare these two implementations of Householder BLR-QR with a block-column-wise Modified Gram-Schmidt (MGS) BLR-QR using fork-join parallelism, and a state-of-the-art vendor-optimized dense Householder QR in Intel MKL. For a matrix of size 131k $\times$ 65k, all BLR methods are more than an order of magnitude faster than the dense QR in MKL. Our methods are also robust to ill-conditioning and produce better orthogonal factors than the existing MGS-based method. On a CPU with 64 cores, our parallel tiled Householder and block-column-wise Householder algorithms show a speedup of 50 and 37 times, respectively.