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

Laura Grigori 1 Sébastien Cayrols 1 James Demmel 2
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : n 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 accuracies, our approach is faster.
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Submitted on : Tuesday, January 1, 2019 - 6:42:36 PM
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Laura Grigori, Sébastien Cayrols, James Demmel. Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2018, 40 (2), pp.C181-C209. ⟨hal-01967901⟩



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