Hutch++:最优随机轨迹估计。

Raphael A Meyer, Cameron Musco, Christopher Musco, David P Woodruff
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引用次数: 73

摘要

我们研究了矩阵a的迹估计问题,该矩阵a只能通过矩阵-向量乘法来访问。本文介绍了一种新的随机化算法Hutch++,该算法仅使用O(1/ε)矩阵向量积计算任何正半定(PSD) a对tr(a)的(1±ε)近似。这改进了无所不在的Hutchinson估计,它需要O(1/ε 2)个矩阵向量积。我们的方法基于一种简单的技术,使用低秩近似步骤来减少Hutchinson估计器的方差,并且易于实现和分析。此外,我们证明了,在所有矩阵向量查询算法中,即使查询可以自适应选择,Hutch++的复杂度在一个对数因子上是最优的。我们在实验中表明,它明显优于Hutchinson的方法。虽然我们的理论要求A是正半定的,但经验增益扩展到涉及非psd矩阵的应用,例如网络中的三角形估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Hutch++: Optimal Stochastic Trace Estimation.

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 ± ε) approximation to tr( A ) for any positive semidefinite (PSD) A using just O(1) matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires O(1 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 requires A to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.

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