Improved Variants of the Hutch++ Algorithm for Trace Estimation

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