Improved variants of the Hutch++ algorithm for trace estimation
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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(ε^-1) matrix-vector products to approximate the trace within a relative error ε with high probability. This compares favorably with the O(ε^-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 symmetric positive semi-definite matrix, we present another variant of Hutch++, called Nyström++, which utilizes the so called Nyström approximation and requires only one pass over the matrix, as compared to two passes with Hutch++. We extend the analysis of Hutch++ to Nyström++. Numerical experiments demonstrate the effectiveness of our two new algorithms.
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