Positive definite matrices and the Symmetric Stein Divergence

Positive definite matrices abound in a dazzling variety of applications. This dazzle could in part be attributed to their rich geometric structure: they form a self-dual convex cone whose strict interior is a Riemannian (also Finslerian) manifold. The manifold view comes with a "natural" distance function while the conic view does not. Nevertheless, drawing motivation from the convex conic view, we introduce the \emph{S-Divergence} as a "natural" distance-like function on the open cone of positive definite matrices. We motivate this divergence via a sequence of results that connect it to the Riemannian metric. In particular, we show that (a) this divergence is the square of a metric; and (b) that it has several geometric properties in common with the Riemannian metric, without being numerically as burdensome. The S-Divergence is even more intriguing: although nonconvex, we show that one can still solve multivariable matrix means using it to global optimality. We complement our results by listing some open problems.