Fast estimation of tr(f(A)) via stochastic Lanczos quadrature

The problem of estimating the trace of matrix functions appears in applications ranging from machine learning and scientific computing, to computational biology. This paper presents an inexpensive method to estimate the trace of f(A) for cases where f is analytic inside a closed interval and A is a symmetric positive definite matrix. The method combines three key ingredients, namely, the stochastic trace estimator, Gaussian quadrature, and the Lanczos algorithm. As examples, we consider the problems of estimating the log-determinant (f(t) = log(t)), the Schatten p-norms (f(t) = t^p/2), the Estrada index (f(t) = e^t), and the trace of the matrix inverse (f(t) = t^-1). We establish multiplicative and additive error bounds for the approximations obtained by this method. In addition, we present error bounds for other useful tools such as approximating the log-likelihood function in the context of maximum likelihood estimation of Gaussian processes. Numerical experiments illustrate the performance of the proposed method on di erent problems arising from various applications.