Subspace-Orbit Randomized Decomposition for Low-rank Matrix Approximation

An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed Subspace-Orbit Randomized singular value decomposition (SORSVD), which makes use of random sampling techniques to give an approximation to a low-rank matrix. Given a large and dense data matrix of size m × n with numerical rank k, where k min{m,n}, the algorithm requires a few passes through data, and can be computed in O(mnk) floating-point operations. Moreover, the SOR-SVD algorithm can utilize advanced computer architectures, and, as a result, it can be optimized for maximum efficiency. The SOR-SVD algorithm is simple, accurate, and provably correct, and outperforms previously reported techniques in terms of accuracy and efficiency. Our numerical experiments support these claims.