On bounding the Thompson metric by Schatten norms.

The 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 d T units, in the Thompson metric, of the actual solution 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 such an upper bound, namely that ∥ X − Y ∥ p ≤ 2 1 p e d − 1 e d max ∥ X ∥ p , ∥ Y ∥ p where ⋅ p denotes the Schatten p-norm and d denotes the Thompson metric between X and Y. Furthermore, a more geometric proof leads to a slightly better bound in the case of the Frobenius norm, ∥ X − Y ∥ 2 ≤ e d − 1 e 2 d + 1 ∥ X ∥ 2 2 + ∥ Y ∥ 2 2 ≤ 2 1 2 e d − 1 e 2 d + 1 max ∥ X ∥ p , ∥ Y ∥ p . [ABSTRACT FROM AUTHOR]