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.

从锥形视角出发，引入 S-Divergence 作为正定矩阵开锥上 ' 自然 ' 距离函数，表现出类似于 Riemannian distance 的几何特征，并通过实验验证了其可用于计算矩阵中位数的优化算法。