Tight Computationally Efficient Approximation of Matrix Norms with Applications

TL;DR

This paper develops efficient methods for approximating matrix norms within specific norm families, enabling applications in control, signal recovery, and system identification despite the NP-hardness of exact computation.

Contribution

It introduces computationally efficient upper bounds for operator norms under ellitopic and co-ellitopic norms, extending to uncertain matrices and practical applications.

Findings

Efficient norm approximation algorithms for ellitopic and co-ellitopic norms.

Bounding robust operator norms for uncertain matrices.

Applications in control synthesis, signal recovery, and system identification.

Abstract

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of "solvable cases," most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space ("ellitopic" and "co-ellitopic," respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding "robust operator norm of uncertain matrix with box uncertainty," that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of "uncertain-but-bounded" noise varying in a box.