Positive definite matrices and the S-divergence

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function while the conic view is not. Nevertheless, drawing motivation from the conic view, we introduce the S-Divergence as a "natural" distance-like function on the open cone of positive definite matrices. We motivate the S-divergence via a sequence of results that connect it to the Riemannian distance. In particular, we show that (a) this divergence is the square of a distance; and (b) that it has several geometric properties similar to those of the Riemannian distance, though without being computationally as demanding. The S-divergence is even more intriguing: although nonconvex, we can still compute matrix means and medians using it to global optimality. We complement our results with some numerical experiments illustrating our theorems and our optimization algorithm for computing matrix medians.