Randomized LU decomposition using sparse projections

Abstract

A fast algorithm for the approximation of a low rank LU decomposition is presented. In order to achieve a low complexity, the algorithm uses sparse random projections combined with FFTbased random projections. The asymptotic approximation error of the algorithm is analyzed and a theoretical error bound is presented. Finally, numerical examples illustrate that for a similar approximation error, the sparse LU algorithm is faster than recent state-of-the-art methods. The algorithm is completely parallelizable that enables to run on a GPU. The performance is tested on a GPU card, showing a significant improvement in the running time in comparison to sequential execution.

Matrix decompositions using sub-Gaussian random matrices

“…A new algorithm is presented, which yields with high probability, a rank r SVD approximation for an m × n matrix that achieves an asymptotic complexity of O(nnz(A)pk + (m + n)k 2 ). Additionally, we showed that the approximated LU algorithm in [1], which uses sub-Gaussian random matrices, has a computational complexity of O(nnz(A)pk + (m + n)k 2 ). We showed in the experiments that although the derived error bounds are not as tight as the bounds from the algorithms in [3,9], in practice, the algorithm in this paper reaches the same error in less time.…”

“…This algorithm does not take advantage of the fact that Ω can be a sparse matrix. Thus, Algorithm 5.1 can be adapted similarly to the algorithm in Theorem 47 [3] and to the LU decomposition algorithm [1]. For SVD approximation to be of rank r, we use the following version of Weyl's inequality: Proof.…”

“…A sub-Gaussian distribution can be used instead of the sparse embedding matrix distribution. Since the correctness proof of the algorithm in [1] is based on Theorem 3.1, it is also applicable for sub-Gaussian matrices.…”

“…The randomized LU decomposition algorithm in [1] is based on the ideas from [3]. We show in Section 4.2 that it is also valid when random matrices from a sub-Gaussian distribution are chosen with the complexity and error bound equal to those from the SVD decomposition.…”

“…The paper has the following structure: In Section 2, we present the necessary mathematical preliminaries. In Section 3, we show that i.i.d sub-Gaussian random matrices are metric conserving and in Section 4 we describe the SVD algorithm and show that the LU algorithm in [1] is valid with i.i.d sub-Gaussian random matrices. In section 5, we present the numerical results of the described SVD algorithm.…”