We consider the problem of estimating the trace of a matrix function $f(A)$. In certain situations, in particular if $f(A)$ cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from [E. Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also characterize situations in which using just one stochastic vector is always -- not only in expectation -- better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods.

翻译：我们考虑矩阵函数$f(A)$迹的估计问题。在某些情况下，特别是当$f(A)$无法通过低秩矩阵良好逼近时，将基于图着色的探测方法与随机迹估计技术相结合，能够在适中计算成本下获得精确近似值。然而，此前这类方法尚未得到深入分析，仅被实践者作为高效启发式算法使用。本文对随机探测方法进行详细分析，重点揭示在何种条件下，随机探测方法中期望近似误差随矩阵维度的增长比例优于非随机探测方法。通过拓展文献[E. Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014]的研究结果，我们还刻画了仅使用单个随机向量总是优于（而不仅限于期望意义下）确定性探测方法的情形。多项数值实验验证了我们的理论分析，并与现有方法进行了比较。