Monte Carlo estimators for the Schatten p-norm of symmetric positive semidefinite matrices

We present numerical methods for computing the Schatten p-norm of positive semi-definite matrices. Our motivation stems from uncertainty quantification and optimal experimental design for inverse problems, where the Schatten p-norm defines a measure of uncertainty. Computing the Schatten p-norm of high-dimensional matrices is computationally expensive. We propose a matrix-free method to estimate the Schatten p-norm using a Monte Carlo estimator and derive convergence results and error estimates for the estimator. To efficiently compute the Schatten p-norm for non-integer and large values of p, we use an estimator using Chebyshev polynomial approximations and extend our convergence and error analysis to this setting as well. We demonstrate the performance of our proposed estimators on several test matrices and in an application to optimal experimental design for a model inverse problem. Key words. Schatten p-norm, Monte Carlo estimator, optimal experimental design, Chebyshev polynomials. AMS subject classifications. 65F35, 65F50, 65C05,