The Generalized Matrix Norm Problem

TL;DR

This paper investigates the computational complexity of the generalized matrix norm problem, identifying cases where it is solvable efficiently and exploring approximation methods and duality principles for broader scenarios.

Contribution

It introduces a comprehensive analysis of the generalized matrix norm problem, including polynomial-time solutions, approximation strategies, and new duality principles involving push-forward and pull-back of seminorms.

Findings

Polynomial-time solvability in certain cases

Development of approximation techniques for complex cases

Discovery of novel duality principles in norm optimization

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 exactly 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.