On bounding the Thompson metric by Schatten norms

David A Snyder,Hari M Srivastava

doi:10.1080/25742558.2019.1614318

David A Snyder, Hari M Srivastava

Open Access

https://doi.org/10.1080/25742558.2019.1614318

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Abstract

AbstractThe Thompson metric provides key geometric insights in the study of non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within dT...

Highlights

The Thompson metric is a variant of the Hilbert metric (Nussbaum & Walsh, 2004)
While the proofs presented in this paper do not explicitly require the matrices be positive definite, in such cases the Thompson metric may be infinite, when the matrix is not positive definite the results presented here are trivial as any finite metric is 1
Proofs not based on the matrix structure of X and Y but based purely on the ordering (Löwner ordering in this case) and norm (Schatten p-norm) being compared might allow for tighter bounds on k X À Ykp even in the absence of any knowledge of the spectrum of X, other than perhaps a restriction that X and Y be positive semidefinite

Summary

Introduction

The Thompson metric is a variant of the Hilbert metric (Nussbaum & Walsh, 2004). The Hilbert metric generalizes the metric structure of hyperbolic geometry to the generalized concept of cones used in the study of Banach (complete normed vector) spaces, such as the space of Hermitian matrices. Knowing that the solution of a problem X and its nth approximation Xn are dT units apart in the Thompson metric provides little indication of how close Xn is to X, i.e. knowing that Xn αX and X αXn in the Löwner ordering (Baksalary & Pukelsheim, 1991), where α 1⁄4 edT , does not intuitively bound k X À Xn k for any of the usual matrix norms k Á k It is k X À Xn k in a suitable matrix norm, not dT, or similar expressions relating X and Xn in the Löwner ordering, that provides insight as to the quality of an approximation Xn. In particular, considering the matrices Xn and X as linear operators on Euclidean vector spaces, the spectral norm, i.e. a Schatten p-norm with p 1⁄4 1, of X À Xn is the relevant measure of how well Xn approximates X. This paper will serve as the beginning of a conversation leading to ever tighter bounds on k X À Ykp given d 1⁄4 dTðX; YÞ as well as minimal information about X and Y, such as their norms and perhaps some knowledge of their spectra of eigenvalues

Preliminaries

Numerical results

Discussion