New Results on CGR/CUR Approximation of a Matrix

TL;DR

This paper introduces accelerated algorithms for CUR and low-rank matrix approximations, demonstrating their efficiency on real-world data and exploring potential improvements through pre-processing techniques.

Contribution

The paper presents new fast algorithms for CUR and low-rank approximations, including acceleration techniques and analysis of hard input classes, with extensions to other computational methods.

Findings

Consistently close CUR approximations on real-world data

Significant reduction in computational cost

Potential for further improvements via pre-processing

Abstract

CUR and low-rank approximations are among most fundamental subjects of numerical linear algebra, with a wide range of applications to a variety of highly important areas of modern computing, which range from the machine learning theory and neural networks to data mining and analysis. We first dramatically accelerate computation of such approximations for the average input matrix, then show some narrow classes of hard inputs for our algorithms, and finally point out a tentative direction to narrowing such classes further by means of pre-processing with quasi Gaussian structured multipliers. Our extensive numerical tests with a variety of real world inputs for regularization from Singular Matrix Database have consistently produced reasonably close CUR approximations at a low computational cost. There is a variety of efficient applications of our results and our techniques to important subjects of matrix computations. Our study provides new insights and enable design of faster algorithms for low-rank approximation by means of sampling and oversampling, for Gaussian elimination with no pivoting and block Gaussian elimination, and for the approximation of railing singular spaces associated with the $\nu$ smallest singular values of a matrix having numerical nullity. We conclude the paper with novel extensions of our acceleration to the Fast Multipole Method and the Conjugate Gradient Algorithms. (This report is an outline of our paper in arXiv:1611.01391.)