Low-rank approximation of large-scale matrices via randomized methods

摘要

Decomposition of a matrix into low-rank matrices is a powerful tool for scientific computing and data analysis. The purpose is to obtain a low-rank matrix by decomposition of the original matrix into a product of smaller and lower-rank matrices or by randomly projecting the matrix down to a lower-dimensional space. Such decomposition requires less storage and computational burden. The focus of this paper is on randomized methods which try as much as possible to preserve the original matrix properties by applying the subspace sampling. In many applications, randomized algorithms in terms of accuracy, stability and speed are much better than the classical decomposition algorithms. In this study, we propose a sparse orthogonal transformation matrix to reduce the dimension of the data. The results show that compared with the most accurate methods, the transformation speed is much faster and can save a lot of memory in the case of huge matrices.