Structure-Aware Analyses and Algorithms for Interpolative Decompositions

Abstract: Low-rank approximation is a task of critical importance in modern science, engineering, and statistics. Many low-rank approximation algorithms, such as the randomized singular value decomposition (RSVD), project their input matrix into a subspace approximating the span of its leading singular vectors. Other algorithms compress their input into a small subset of representative rows or columns, leading to a so-called interpolative decomposition. This paper investigates how the accuracy of interpolative decompositions is affected by the structural properties of the input matrix being operated on, including how these properties affect the performance comparison between interpolative decompositions and RSVD. We also introduce a novel method of interpolative decomposition in the form of the randomized Golub-Klema-Stewart (RGKS) algorithm, which combines RSVD with a pivoting strategy for column subset selection. Through numerical experiments, we find that matrix structures including singular subspace geometry and singular spectrum decay play a significant role in determining the performance comparison between these different algorithms. We also prove inequalities which bound the error of a general interpolative decomposition in terms of these matrix structures. Lastly, we develop forms of these bounds specialized to RGKS while considering how randomization affects the approximation error of this algorithm.