Efficient algorithms for cur and interpolative matrix decompositions

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation. For certain applications, further acceleration can be attained by incorporating techniques based on randomized projections. Numerical experiments demonstrate advantageous performance compared to existing techniques for computing CUR factorizations.

[1]  Matemática,et al.  Society for Industrial and Applied Mathematics , 2010 .

[2]  T. Chan.  Rank revealing QR factorizations , 1987 .

[3]  Per-Gunnar Martinsson,et al.  RSVDPACK: Subroutines for computing partial singular value decompositions via randomized sampling on single core, multi core, and GPU architectures , 2015, ArXiv.

[4]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[5]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[6]  Petros Drineas,et al.  CUR matrix decompositions for improved data analysis , 2009, Proceedings of the National Academy of Sciences.

[7]  V. Rokhlin,et al.  A randomized algorithm for the approximation of matrices , 2006 .

[8]  Ming Gu,et al.  Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization , 1996, SIAM J. Sci. Comput..

[9]  R. Welsch,et al.  The Hat Matrix in Regression and ANOVA , 1978 .

[10]  Christos Boutsidis,et al.  Optimal CUR matrix decompositions , 2014, STOC.

[11]  Danny C. Sorensen,et al.  A DEIM Induced CUR Factorization , 2014, SIAM J. Sci. Comput..

[12]  Per-Gunnar Martinsson,et al.  RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures , 2015 .

[13]  Muhammad Tayyab Asif,et al.  CUR decomposition for compression and compressed sensing of large-scale traffic data , 2013, 16th International IEEE Conference on Intelligent Transportation Systems (ITSC 2013).

[14]  Eugene E. Tyrtyshnikov,et al.  Incomplete Cross Approximation in the Mosaic-Skeleton Method , 2000, Computing.

[15]  Mark Tygert,et al.  A Randomized Algorithm for Principal Component Analysis , 2008, SIAM J. Matrix Anal. Appl..

[16]  Michael W. Mahoney,et al.  rCUR: an R package for CUR matrix decomposition , 2012, BMC Bioinformatics.

[17]  S. Goreinov,et al.  A Theory of Pseudoskeleton Approximations , 1997 .

[18]  S. Muthukrishnan,et al.  Relative-Error CUR Matrix Decompositions , 2007, SIAM J. Matrix Anal. Appl..

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Per-Gunnar Martinsson,et al.  Randomized algorithms for the low-rank approximation of matrices , 2007, Proceedings of the National Academy of Sciences.

[21]  Tamás Sarlós,et al.  Improved Approximation Algorithms for Large Matrices via Random Projections , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[22]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[23]  V. Rokhlin,et al.  A fast randomized algorithm for the approximation of matrices ✩ , 2007 .

[24]  Zhihua Zhang,et al.  Improving CUR matrix decomposition and the Nyström approximation via adaptive sampling , 2013, J. Mach. Learn. Res..