Positive definite matrices and the S-divergence

Hermitian positive definite (hpd) matrices form a self-dual convex cone whose interior is a Riemannian manifold of nonpositive curvature. The manifold view comes with a natural distance function but the conic view does not. Thus, drawing motivation from convex optimization we introduce the S-divergence, a distance-like function on the cone of hpd matrices. We study basic properties of the S-divergence and explore its connections to the Riemannian distance. In particular, we show that (i) its square-root is a distance, and (ii) it exhibits numerous nonpositive-curvature-like properties.