Improved Variants of the Hutch++ Algorithm for Trace Estimation

This paper is concerned with two improved variants of the Hutch++ algorithm for estimating trace a square matrix, implicitly given through matrix-vector products. combines randomized low-rank approximation in first phase stochastic estimation second phase. In turn, only requires $O\left(\varepsilon^{-1}\right)$ products to approximate within relative error $\varepsilon$ high probability, provided that matrix symmetric positive semidefinite. compares favorably $O\left(\varepsilon^{-2}\right)$ needed when using alone. Hutch++, number fixed priori and distributed prescribed fashion among phases. this work, we derive an adaptive variant which outputs estimate some tolerance controllable failure while splitting near-optimal way For special case semidefinite present another called Nyström++, utilizes so-called Nyström one pass over as compared passes Hutch++. We extend analysis Nyström++. Numerical experiments demonstrate effectiveness our new algorithms.