Superfast CUR Matrix Algorithms, Their Pre-Processing and Extensions

TL;DR

This paper investigates superfast algorithms for low-rank matrix approximation, demonstrating their effectiveness on various matrix classes, analyzing their limitations, and proposing new techniques for improved performance and extensions to other algorithms.

Contribution

The paper introduces new theoretical insights into the performance of superfast LRA algorithms, including their success on broad matrix classes and novel pre-processing methods.

Findings

Superfast algorithms compute accurate LRAs for most matrices with proper pre-processing.

Certain hard input matrices cause failure of some superfast algorithms, but these are rare.

Empirical results show accuracy is maintained with sparse, structured multipliers.

Abstract

We study superfast algorithms that computes low rank approximation of a matrix (hereafter referred to as LRA) that use much fewer memory cells and arithmetic operations than the input matrix has entries. We first specify a family of 2mn matrices of size m*n such that for almost 50% of them any superfast LRA algorithm fails to improve the poor trivial approximation by the matrix filled with zeros, but then we prove that the class of all such hard inputs is narrow - the cross-approximation (hereafter {C-A}) superfast iterations as well as some more primitive superfast algorithms compute reasonably accurate LRAs in their transparent CUR form (i) to any matrix allowing close LRA except for small norm perturbations of matrices of an algebraic variety of a smaller dimension, (ii) to the average matrix allowing close LRA, (iii) to the average sparse matrix allowing close LRA and (iv) with a high probability to any matrix allowing close LRA if it is pre-processed fast with a random Gaussian, SRHT or SRFT multiplier. Moreover empirically the output LRAs remain accurate when we perform the computations superfast by replacing such a multiplier with one of our sparse and structured multipliers. Our techniques, auxiliary results and extensions may be of some independent interest. We analyze C-A and other superfast algorithms twice -- based on two well-known sufficient criteria for obtaining accurate LRAs. We provide a distinct proof in the case of superfast variant of randomized algorithms of [DMM08], improve a decade-old estimate for the norm of the inverse of a Gaussian matrix, prove such an estimate also in the case of a sparse Gaussian matrix, present some novel advanced pre-processing techniques for fast and superfast computation of LRA, and extend our results to dramatic acceleration of the Fast Multipole Method (FMM) and the Conjugate Gradient algorithms.