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arXiv · 2010.09649

Hutch++: Optimal Stochastic Trace Estimation

Raphael A. Meyer·Cameron Musco·Christopher Musco·David P. Woodruff
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Abstract

We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \pm \epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/\epsilon)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/\epsilon^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.

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BibTeXRIS

Raphael A. Meyer, Cameron Musco, Christopher Musco, David P. Woodruff. 2020-10-19. Hutch++: Optimal Stochastic Trace Estimation. https://arxiv.org/abs/2010.09649

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