Stochastic trace estimation for positive trace-class operators
Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\mathcal{O}(\varepsilon^{-2})$ products to achieve accuracy $\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\mathcal{O}(\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\mathcal{O}(\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning&Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.