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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 alow-rank matrix, combining probing methods based on graph colorings withstochastic trace estimation techniques can yield accurate approximations atmoderate cost. So far, such methods have not been thoroughly analyzed, though,but were rather used as efficient heuristics by practitioners. In thismanuscript, we perform a detailed analysis of stochastic probing methods and,in particular, expose conditions under which the expected approximation errorin the stochastic probing method scales more favorably with the dimension ofthe matrix than the error in non-stochastic probing. Extending results from [E.Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensionalGaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we alsocharacterize 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 existingmethods.

Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices