On the geometry of covariance matrices

We introduce and compare certain distance measures between covariance matrices. These originate in information theory, quantum mechanics and optimal transport. More specifically, we show that the Bures/Hellinger distance between covariance matrices coincides with the Wasserstein-2 distance between the corresponding Gaussian distributions. We also note that this Bures/Hellinger/Wasserstein distance can be expressed as the solution to a linear matrix inequality (LMI). A consequence of this fact is that the computational cost in covariance approximation problems scales nicely with the size of the matrices involved. We discuss the relevance of this metric in spectral-line detection and spectral morphing.