Sublinear Randomized Algorithms for Skeleton Decompositions

Abstract

A skeleton decomposition of a matrix A is any factorization of the form A[subscript :C]ZA[subscript R:], where A[subscript :C] comprises columns of A, and A[subscript R:] comprises rows of A. In this paper, we investigate the conditions under which random sampling of C and R results in accurate skeleton decompositions. When the singular vectors (or more generally the generating vectors) are incoherent, we show that a simple algorithm returns an accurate skeleton in sublinear O(ℓ[superscript 3]) time from ℓ ~ k log n rows and columns drawn uniformly at random, with an approximation error of the form O([n over ℓ]σ[subscript k]) whereσ[subscript k] is the kth singular value of A. We discuss the crucial role that regularization plays in forming the middle matrix U as a pseudoinverse of the restriction A[subscript RC] of A to rows in R and columns in C. The proof methods enable the analysis of two alternative sublinear-time algorithms, based on the rank-revealing QR decomposition, which allows us to tighten the number of rows and/or columns to k with error bound proportional to σ[subscript k].