Fisher-Rao distance and pullback SPD cone distances between multivariate normal distributions

Data sets of multivariate normal distributions abound in many scientific areas like diffusion tensor imaging, structure tensor computer vision, radar signal processing, machine learning, just to name a few. In order to process those normal data sets for downstream tasks like filtering, classification or clustering, one needs to define proper notions of dissimilarities between normals and paths joining them. The Fisher-Rao distance defined as the Riemannian geodesic distance induced by the Fisher information metric is such a principled metric distance which however is not known in closed-form excepts for a few particular cases. In this work, we first report a fast and robust method to approximate arbitrarily finely the Fisher-Rao distance between multivariate normal distributions. Second, we introduce a class of distances based on diffeomorphic embeddings of the normal manifold into a submanifold of the higher-dimensional symmetric positive-definite cone corresponding to the manifold of centered normal distributions. We show that the projective Hilbert distance on the cone yields a metric on the embedded normal submanifold and we pullback that cone distance with its associated straight line Hilbert cone geodesics to obtain a distance and smooth paths between normal distributions. Compared to the Fisher-Rao distance approximation, the pullback Hilbert cone distance is computationally light since it requires to compute only the extreme minimal and maximal eigenvalues of matrices. Finally, we show how to use those distances in clustering tasks.