ContHutch++: Stochastic trace estimation for implicit integral operators

TL;DR

A generalization of Hutch++ is proposed, which is called ContHutch++, that uses operator-function products to efficiently estimate the trace of any trace-class integral operator to avoid spectral artifacts introduced by discretization and are accompanied by rigorous high-probability error bounds.

Abstract

Hutchinson's estimator is a randomized algorithm that computes an $ε$-approximation to the trace of any positive semidefinite matrix using $\mathcal{O}(1/ε^2)$ matrix-vector products. An improvement of Hutchinson's estimator, known as Hutch++, only requires $\mathcal{O}(1/ε)$ matrix-vector products. In this paper, we propose a generalization of Hutch++, which we call ContHutch++, that uses operator-function products to efficiently estimate the trace of any trace-class integral operator. Our ContHutch++ estimates avoid spectral artifacts introduced by discretization and are accompanied by rigorous high-probability error bounds. We use ContHutch++ to derive a new high-order accurate algorithm for quantum density-of-states and also show how it can estimate electromagnetic fields induced by incoherent sources.

Figures (10)

• Figure 1: \newlabelfig:toy_example10 Two symmetric positive definite kernels are used for trace estimation experiments: the Helmholtz-like kernel in \ref{['eqn:kernel1']} (left panel) and the combination of sinc kernels in \ref{['eqn:kernel2']} (right panel).
• Figure 1: Domains of integration for computing the second term in \ref{['eq:bound2']}.
• Figure 1: \newlabelfig:rsvd_scaling0 In the left panel, the constant $\eta$ in \ref{['cor:ContHutch++']} is calculated for the trace-class kernel in \ref{['eqn:kernel2']} and plotted against $k$ (the total number of probe vectors is $k+p=2k$) used in the continuous randomized range finder for $\ell=0.04$ (blue), $\ell=0.02$ (red), and $\ell=0.01$ (yellow) in the squared exponential covariance kernel $K_{\rm SE}$. In the right panel, the corresponding relative error bound for the rangefinder is plotted against $k$ for $\ell=0.04$ (blue), $\ell=0.02$ (red), and $\ell=0.01$ (yellow) and compared with the best low-rank approximation error (black).
• Figure 1: \newlabelfig:dos10 The first six rational convolution kernels based on equispaced points are plotted in the left panel. The eigenvalues of the corresponding operators in \ref{['eqn:dos2']} with $\mathcal{L} = -\Delta$, $\lambda=10$, and $\sigma = 0.25$, are plotted in the right panel.
• Figure 2: \newlabelfig:toy_example1_conv0 The continuous analog of Hutchinson's estimator converges in mean up to a limiting bias determined by the covariance parameter $\ell>0$ for the Gaussian process, as demonstrated for the Helmholtz-like kernel in \ref{['eqn:kernel1']} (left panel). This limiting bias decreases at a controlled rate as $\ell\rightarrow 0$ when $m$ is selected adaptively to balance the sample error and covariance error (right panel). The precise rate of decrease depends on the kernel's modulus of continuity (see \ref{['thm:expectation1']}).

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 1
• Proof 1
• Theorem 2
• Proof 2
• Corollary 3
• Proof 3
• Theorem 4
• Proof 4