Randomized LU decomposition using sparse projections

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 FFT-based 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.

[1]  Huy L. Nguyen,et al.  Lower Bounds for Oblivious Subspace Embeddings , 2013, ICALP.

[2]  David P. Woodruff,et al.  Low rank approximation and regression in input sparsity time , 2012, STOC '13.

[3]  David P. Woodruff,et al.  Low rank approximation and regression in input sparsity time , 2013, STOC '13.

[4]  Petros Drineas,et al.  FAST MONTE CARLO ALGORITHMS FOR MATRICES II: COMPUTING A LOW-RANK APPROXIMATION TO A MATRIX∗ , 2004 .

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

[6]  M. Gu,et al.  Strong rank revealing LU factorizations , 2003 .

[7]  Daniel M. Kane,et al.  Sparser Johnson-Lindenstrauss Transforms , 2010, JACM.

[8]  Nathan Halko,et al.  An Algorithm for the Principal Component Analysis of Large Data Sets , 2010, SIAM J. Sci. Comput..

[9]  Bernard Chazelle,et al.  The Fast Johnson--Lindenstrauss Transform and Approximate Nearest Neighbors , 2009, SIAM J. Comput..

[10]  Dimitris Achlioptas,et al.  Fast computation of low rank matrix approximations , 2001, STOC '01.

[11]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[12]  C. Chui,et al.  Article in Press Applied and Computational Harmonic Analysis a Randomized Algorithm for the Decomposition of Matrices , 2022 .

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

[14]  Huy L. Nguyen,et al.  OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings , 2012, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[15]  Amir Averbuch,et al.  Randomized LU Decomposition , 2013, ArXiv.

[16]  Joel A. Tropp,et al.  Improved Analysis of the subsampled Randomized Hadamard Transform , 2010, Adv. Data Sci. Adapt. Anal..

[17]  Mark Tygert,et al.  An implementation of a randomized algorithm for principal component analysis , 2014, ArXiv.

[18]  Huy L. Nguyen,et al.  Sparsity lower bounds for dimensionality reducing maps , 2012, STOC '13.

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

[20]  Martin Raab,et al.  "Balls into Bins" - A Simple and Tight Analysis , 1998, RANDOM.

[21]  C. Pan On the existence and computation of rank-revealing LU factorizations , 2000 .

[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 .