  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.
