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We present randomized algorithms for estimating the trace and deter- minantof Hermitian positive semi-definite matrices. The algorithms are based onsubspace iteration, and access the matrix only through matrix vector products.We analyse the error due to randomization, for starting guesses whose elementsare Gaussian or Rademacher random variables. The analysis is cleanly separatedinto a structural (deterministic) part followed by a probabilistic part. Ourabsolute bounds for the expectation and concentration of the estimators arenon-asymptotic and informative even for matrices of low dimension. For thetrace estimators, we also present asymptotic bounds on the number of samples(columns of the starting guess) required to achieve a user-specified relativeerror. Numerical experiments illustrate the performance of the estimators andthe tightness of the bounds on low-dimensional matrices; and on a challengingapplication in uncertainty quantification arising from Bayesian optimalexperimental design.

Randomized Matrix-free Trace and Log-Determinant Estimators