Efficient randomized algorithms for adaptive low-rank factorizations of large matrices

In this paper, randomized techniques for computing low-rank factorizations are presented. The proposed methods take in a tolerance $\varepsilon$ and an $m \times n$ matrix $\boldsymbol{A}$, and output an approximate low-rank factorization of $\boldsymbol{A}$, whose error measured in the Frobenius norm is within $\varepsilon$. The techniques are based on the blocked randQB scheme proposed by P.-G. Martinsson and S. Voronin, producing a QB factorization. By employing an economic error indicator and moving $\boldsymbol{A}$ out of the loop, the techniques result in two algorithms called randQB_EI and randQB_FP. They are mathematically equivalent to the existing blocked scheme, but are more computationally efficient. The randQB_FP algorithm also owns the merit of pass-efficiency. Numerical experiments on a multi-core parallel computing server show that the proposed algorithms have the similar accuracy as the blocked randQB scheme, but cost a small fraction of runtime and memory. The benefits are even larger (up to 20X) for handling large sparse matrices. Compared with the adaptive range finder algorithm, the proposed methods output much smaller and close to optimal rank while satisfying the preset accuracy tolerance.