Sublinear Randomized Algorithms for Skeleton Decompositions

Abstract

A skeleton decomposition of a matrix $A$ is any factorization of the form $A_{:C} Z A_{R:}$, where $A_{:C}$ comprises columns of $A$, and $A_{R:}$ 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 $O(\ell^3)$ time from $\ell \simeq k \log n$ rows and columns drawn uniformly at random, with an approximation error of the form $O(\frac{n}{\ell} \sigma_k)$ where $\sigma_k$ is the $k$th singular value of $A$. We discuss the crucial role that regularization plays in forming the middle matrix $U$ as a pseudoinverse of the restriction $A_{RC}$ 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 $\sigma_k$.