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

Electronic Transactions on Numerical Analysis

article

Ethan Dudley¹  Arvind K. Saibaba¹  Alen Alexanderian¹ 

[1] Department of Mathematics, North Carolina State University;Department of Mathematics, University of Maryland, College Park

关键词: Schatten $p$-norm;    Monte Carlo estimator;    optimal experimental design;    Chebyshev polynomials.;   

DOI  :  10.1553/etna_vol55s213

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

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.

【 授权许可】

Unknown