Monte Carlo Estimators for the Schatten p-norm of Symmetric Positive Semidefinite Matrices

TL;DR

This paper introduces Monte Carlo and polynomial approximation methods for efficiently estimating the Schatten p-norm of large positive semi-definite matrices, with applications in uncertainty quantification and experimental design.

Contribution

It develops a matrix-free Monte Carlo estimator for the Schatten p-norm, including convergence analysis and error bounds, applicable to non-integer and large p values.

Findings

Estimator performs well on test matrices

Method is effective for high-dimensional matrices

Application improves experimental design in inverse problems

Abstract

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 design criterion known as the P-optimal criterion. 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 a Chebyshev polynomial approximation 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 through an application to optimal experimental design of a model inverse problem.