Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting

James Demmel, Laura Grigori and Sebastien Cayrols

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2016-122
June 21, 2016

http://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-122.pdf

In this paper we present an algorithm for computing a low rank approximation of a sparse matrix based on a truncated LU factorization with column and row permutations. We present various approaches for determining the column and row permutations that show a trade-off between speed versus deterministic/probabilistic accuracy. We show that if the permutations are chosen by using tournament pivoting based on QR factorization, then the obtained truncated LU factorization with column/row tournament pivoting, LU CRTP, satisfies bounds on the singular values which have similarities with the ones obtained by a communication avoiding rank revealing QR factorization. Experiments on challenging matrices show that LU CRTP provides a good low rank approximation of the input matrix and it is less expensive than the rank revealing QR factorization in terms of computational and memory usage costs, while also minimizing the communication cost. We also compare the computational complexity of our algorithm with randomized algorithms and show that for sparse matrices and high enough but still modest accuracy, our approach is faster.


BibTeX citation:

@techreport{Demmel:EECS-2016-122,
    Author = {Demmel, James and Grigori, Laura and Cayrols, Sebastien},
    Title = {Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {2016},
    Month = {Jun},
    URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-122.html},
    Number = {UCB/EECS-2016-122},
    Abstract = {In this paper we present an algorithm for computing a low rank approximation of a
sparse matrix based on a truncated LU factorization with column and row permutations. We present
various approaches for determining the column and row permutations that show a trade-off between
speed versus deterministic/probabilistic accuracy. We show that if the permutations are chosen by
using tournament pivoting based on QR factorization, then the obtained truncated LU factorization
with column/row tournament pivoting, LU CRTP, satisfies bounds on the singular values which have
similarities with the ones obtained by a communication avoiding rank revealing QR factorization.
Experiments on challenging matrices show that LU CRTP provides a good low rank approximation
of the input matrix and it is less expensive than the rank revealing QR factorization in terms of
computational and memory usage costs, while also minimizing the communication cost. We also
compare the computational complexity of our algorithm with randomized algorithms and show that
for sparse matrices and high enough but still modest accuracy, our approach is faster.}
}

EndNote citation:

%0 Report
%A Demmel, James
%A Grigori, Laura
%A Cayrols, Sebastien
%T Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting
%I EECS Department, University of California, Berkeley
%D 2016
%8 June 21
%@ UCB/EECS-2016-122
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-122.html
%F Demmel:EECS-2016-122