New Results on CGR/CUR Approximation of a Matrix

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.)