Speaker
Description
The implicit trace estimation is a problem of approximating the trace of a matrix $A$ accessed only through matrix-vector multiplication $x \mapsto Ax$, with the goal of using as few multiplications as possible to obtain an accurate approximation. Girard-Hutchinson's estimator computes the $\varepsilon$-approximation using $\mathcal{O}(\varepsilon^{-2})$ products, while its' variance-reducing improvement, known as Hutch++, only requires $\mathcal{O}(\varepsilon^{-2})$ products. In this talk, we propose an extension of those algorithms to estimate the trace of a trace-class operator by using operator-function multiplications. Our estimator is based on the notion of a Gaussian distribution on an arbitrary Hilbert space, so precise high-probability error bounds can be derived analogously to the matrix case estimators. We compare the performance of our method with an existing algorithm of this type, the ContHutch++ algorithm by Zvonek, Horning & Townsend. The theoretical results we prove concern mostly positive definite operators, but we provide empirical results that show potential in applications to general trace-class operators.