An Iterative Random Sampling Algorithm for Rapid and Scalable Estimation of Matrix Spectra

An easy-to-implement iterative algorithm that enables efficient and scalable spectral analysis of dense matrices is presented. The algorithm relies on the approximation of a matrix's singular values by those of a series of smaller matrices formed from uniform random sampling of its rows and columns. It is shown that, for sufficiently incoherent and rank-deficient matrices, the singular values [are expected to] decay at the same rate as those of matrices formed via this sampling scheme, which permits such matrices’ ranks to be accurately estimated from the smaller matrices’ spectra. Moreover, for such a matrix of size
$m \times n$
, it is shown that the dominant singular values are [expected to be]
$\sqrt {mn} /k$
times those of a
$k \times k$
matrix formed by randomly sampling
$k$
of its rows and columns. Starting from a small initial guess
$k\ = {k}_0$
, the algorithm repeatedly doubles
$k$
until two convergence criteria are met; the criteria to ensure that
$k$
is sufficiently large to estimate the singular values, to the desired accuracy, are presented. The algorithm's properties are analyzed theoretically and its efficacy is studied numerically for small to very-large matrices that result from discretization of integral-equation operators, with various physical kernels common in electromagnetics and acoustics, as well as for artificial matrices of various incoherence and rank-deficiency properties.