Parallel QR Factorization of Block Low-Rank Matrices

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 \(\mathcal{O}(mn)\),...

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Bibliographic Details
Published in:arXiv.org
Main Authors: Apriansyah, M Ridwan, Yokota, Rio
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 12.08.2022
Subjects:
Algorithms
Columnar structure
Complexity
Factorization
Ill-conditioned problems (mathematics)
ISSN:2331-8422
Online Access:Get full text
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Summary: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.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
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ISSN:2331-8422
DOI:10.48550/arxiv.2208.06194