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This is a Quantitative Study study.

October 10, 2025

The Generalized Matrix Norm Problem

AKAdrian KulmburgKarlsruhe Institute of Technology

Key Points

The operator norm of a matrix can be solved in polynomial time under specific conditions, and further techniques are explored.
We find that approximation methods are applicable in various cases, expanding the utility of matrix norms.
Investigation of push-forward and pull-back norms leads to new duality principles in optimization contexts.
This work received the 'SIAM Reproducibility Badge', providing access to code and data for verification.

Abstract

.We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved in polynomial time in certain situations, and we discuss how it can be approximated in other cases. Along the way, we investigate the concept of push-forward and pull-back of seminorms, which leads us to uncover novel duality principles that come into play when optimizing over the unit ball of norms.Reproducibility of computational results.This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://codeocean.com/capsule/8982170/tree and in the supplementary materials (RBcode.zip 43.0MB).Keywordsoperator normpush-forward normpull-back normapproximationMSC codes15A6065F3568Q25

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