Randomized estimation of spectral densities of large matrices made accurate

Abstract

For a large Hermitian matrix $A\in \mathbb {C}^{N\times N}$, it is often the case that the only affordable operation is matrix–vector multiplication. In such case, randomized method is a powerful way to estimate the spectral density (or density of states) of A. However, randomized methods developed so far for estimating spectral densities only extract information from different random vectors independently, and the accuracy is therefore inherently limited to $\mathcal {O}(1/\sqrt{N_{v}})$ where $N_{v}$ is the number of random vectors. In this paper we demonstrate that the “$\mathcal {O}(1/\sqrt{N_{v}})$ barrier” can be overcome by taking advantage of the correlated information of random vectors when properly filtered by polynomials of A. Our method uses the fact that the estimation of the spectral density essentially requires the computation of the trace of a series of matrix functions that are numerically low rank. By repeatedly applying A to the same set of random vectors and taking different linear combination of the results, we can sweep through the entire spectrum of A by building such low rank decomposition at different parts of the spectrum. Under some assumptions, we demonstrate that a robust and efficient implementation of such spectrum sweeping method can compute the spectral density accurately with $\mathcal {O}(N^2)$ computational cost and $\mathcal {O}(N)$ memory cost. Numerical results indicate that the new method can significantly outperform existing randomized methods in terms of accuracy. As an application, we demonstrate a way to accurately compute a trace of a smooth matrix function, by carefully balancing the smoothness of the integrand and the regularized density of states using a deconvolution procedure.