Spectral Condition-Number Estimation of Large Sparse Matrices

TL;DR

This paper introduces a randomized Krylov-subspace method using LSQR to efficiently estimate the spectral condition number of large sparse matrices, even when they are rank deficient or too large for dense SVD.

Contribution

It presents a novel, memory-efficient approach for estimating the smallest singular value and condition number of large sparse matrices using Krylov subspaces and LSQR.

Findings

Accurately estimates condition numbers of various matrices

Effective for large, sparse, and rectangular matrices

Can identify rank deficiency in matrices

Abstract

We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value \sigma_{\min} of A. Our method estimates this value by solving a consistent linear least-squares problem with a known solution using a specific Krylov-subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to \sigma_{\min}. Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running a dense SVD would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR) and it works equally well on square and rectangular matrices.