Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition

Abstract

Given a matrix $\bf A$ with numerical rank $k$ , the two-sided orthogonal decomposition (TSOD) computes a factorization ${\bf A} = {\bf UDV}^T$ , where ${\bf U}$ and ${\bf V}$ are orthogonal, and ${\bf D}$ is (upper/lower) triangular. TSOD is rank-revealing as the middle factor ${\bf D}$ reveals the rank of $\bf A$ . The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the $k$ approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm.