Matrix decompositions using sub-Gaussian random matrices

Abstract

Abstract In recent years, several algorithms which approximate matrix decomposition have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We present a new algorithm, which achieves with high probability a rank-$r$ singular value decomposition (SVD) approximation of an $n \times n$ matrix and derive an error bound that does not depend on the first $r$ singular values. Although the algorithm has an asymptotic complexity similar to state-of-the-art algorithms and the proven error bound is not as tight as the state-of-the-art bound, experiments show that the proposed algorithm is faster in practice while providing the same error rates as those of the state-of-the-art algorithms. We also show that an i.i.d. sub-Gaussian matrix with large probability of having null entries is metric conserving. This result is used in the SVD approximation algorithm, as well as to improve the performance of a previously proposed approximated LU decomposition algorithm.