Fast Derandomized Low-rank Approximation and Extensions

TL;DR

This paper provides formal theoretical support for the empirical success of structured random sampling in low-rank matrix approximation, derandomizes existing algorithms, and extends these techniques to other fundamental matrix computations.

Contribution

It introduces a novel formal framework for structured random sampling, derandomizes and simplifies existing low-rank approximation algorithms, and applies these methods to related matrix computations.

Findings

Formal support for empirical methods established

Algorithms are derandomized and simplified

Numerical tests confirm theoretical results

Abstract

Low-rank approximation of a matrix by means of structured random sampling has been consistently efficient in its extensive empirical studies around the globe, but adequate formal support for this empirical phenomenon has been missing so far. Based on our novel insight into the subject, we provide such an elusive formal support and derandomize and simplify the known numerical algorithms for low-rank approximation and related computations. Our techniques can be applied to some other areas of fundamental matrix computations, in particular to the Least Squares Regression, Gaussian elimination with no pivoting and block Gaussian elimination. Our formal results and our numerical tests are in good accordance with each other.