On the Relation of Schatten Norms and the Thompson Metric

TL;DR

This paper establishes bounds relating the Thompson metric to Schatten norms and Frobenius norm, providing insights into the quality of matrix approximations in non-linear matrix equations.

Contribution

It introduces upper bounds on Schatten and Frobenius norms based on the Thompson metric and Schatten norms, enhancing understanding of approximation quality.

Findings

Bound on Schatten norm of X - Y in terms of Thompson metric and Schatten norms.

Tighter bound on Frobenius norm of X - Y.

Provides theoretical tools for matrix approximation analysis.

Abstract

The Thompson metric provides key geometric insights in the study or non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within d_T units of the actual solution in the Thompson metric provides little insight into how good the approximation is as a matrix or vector approximation. That is, bounding the Thompson metric between an approximate and accurate solution to a problem does not provide obvious bounds either for the spectral or the Frobenius norm, both Schatten norms, of the difference between the approximation and accurate solution. This paper reports an upper bound on the Schatten norm of X - Y related to both the Thompson metric between X and Y and the maximum of their Schatten norms. This paper reports a similar but slightly tighter bound for the Frobenius norm of X - Y.