Reconstructing Householder Vectors from Tall-Skinny QR

The Tall-Skinny QR (TSQR) algorithm is more communication efficient than the standard Householder algorithm for QR decomposition of matrices with many more rows than columns. However, TSQR produces a different representation of the orthogonal factor and therefore requires more software development to support the new representation. Further, implicitly applying the orthogonal factor to the trailing matrix in the context of factoring a square matrix is more complicated and costly than with the Householder representation. We show how to perform TSQR and then reconstruct the Householder vector representation with the same asymptotic communication efficiency and little extra computational cost. We demonstrate the high performance and numerical stability of this algorithm both theoretically and empirically. The new Householder reconstruction algorithm allows us to design more efficient parallel QR algorithms, with significantly lower latency cost compared to Householder QR and lower bandwidth and latency costs compared with Communication-Avoiding QR (CAQR) algorithm. As a result, our final parallel QR algorithm outperforms ScaLAPACK and Elemental implementations of Householder QR and our implementation of CAQR on the Hopper Cray XE6 NERSC system. We also provide algorithmic improvements to the ScaLAPACK and CAQR algorithms.

[1]  Christian H. Bischof,et al.  A Basis-Kernel Representation of Orthogonal Matrices , 1995, SIAM J. Matrix Anal. Appl..

[2]  James Demmel,et al.  Communication Avoiding Rank Revealing QR Factorization with Column Pivoting , 2015, SIAM J. Matrix Anal. Appl..

[3]  C. Puglisi.  Modification of the householder method based on the compact WY representation , 1992 .

[4]  Alexander Tiskin.  Communication-efficient parallel generic pairwise elimination , 2007, Future Gener. Comput. Syst..

[5]  B. Parlett,et al.  Block reflectors: theory and computation , 1988 .

[6]  Thomas Hérault,et al.  QR factorization of tall and skinny matrices in a grid computing environment , 2009, 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS).

[7]  James Demmel,et al.  Communication-optimal Parallel and Sequential QR and LU Factorizations , 2008, SIAM J. Sci. Comput..

[8]  C. Bischof,et al.  On orthogonal block elimination , 1996 .

[9]  Yusaku Yamamoto,et al.  Backward error analysis of the AllReduce algorithm for householder QR decomposition , 2011, Japan Journal of Industrial and Applied Mathematics.

[10]  James Demmel,et al.  Communication-Avoiding QR Decomposition for GPUs , 2011, 2011 IEEE International Parallel & Distributed Processing Symposium.

[11]  C. Loan,et al.  A Storage-Efficient $WY$ Representation for Products of Householder Transformations , 1989 .

[12]  Mark Hoemmen,et al.  A Communication-Avoiding, Hybrid-Parallel, Rank-Revealing Orthogonalization Method , 2011, 2011 IEEE International Parallel & Distributed Processing Symposium.

[13]  Rajeev Thakur,et al.  Optimization of Collective Communication Operations in MPICH , 2005, Int. J. High Perform. Comput. Appl..

[14]  Jack J. Dongarra,et al.  Scalable Tile Communication-Avoiding QR Factorization on Multicore Cluster Systems , 2010, 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis.

[15]  Nicholas J. Higham,et al.  NLEVP: A Collection of Nonlinear Eigenvalue Problems , 2013, TOMS.

[16]  Robert A. van de Geijn,et al.  Collective communication: theory, practice, and experience , 2007, Concurr. Comput. Pract. Exp..

[17]  James Demmel,et al.  Minimizing Communication in Numerical Linear Algebra , 2009, SIAM J. Matrix Anal. Appl..

[18]  Robert A. van de Geijn,et al.  Elemental: A New Framework for Distributed Memory Dense Matrix Computations , 2013, TOMS.

[19]  G. Golub,et al.  Parallel block schemes for large-scale least-squares computations , 1988 .

[20]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[21]  Thomas Hérault,et al.  Hierarchical QR factorization algorithms for multi-core clusters , 2013, Parallel Comput..

[22]  James Demmel,et al.  Communication-Optimal Parallel 2.5D Matrix Multiplication and LU Factorization Algorithms , 2011, Euro-Par.

[23]  A. Farley.  Broadcast Time in Communication Networks , 1980 .