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 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.
2) The square root of S-divergence is proven to be a metric on Sym + d REF .
Positive definite matrices and the S-divergence
arXiv: Functional Analysis
Mathematics