# Distances defined in manifold of symmetric positive definite matrices

There are papers that describe how using Euclidean distance between symmetric positive definite (SPD) matrices is "wrong" because it does not take into account the geometry of the SPD manifold (paper 1, paper 2). These papers propose using geodesic distances such as the log-Euclidean or the affine invariant metrics to properly define the distance between matrices along the manifold. However, if I understand correctly, SPD matrices lie inside a convex set i.e. linear combinations of SPD matrices will be SPD. Intuitively, I don't see the need of defining a distance that goes along the curvature of SPD matrices since the set is dense. Is this correct or am I missing something?

• can you provide references to what you are referring as 'There are papers'? Dec 11 '17 at 19:27
• I had put the references in the first sentence as 1, 2. I made them more noticeable now Dec 11 '17 at 20:29
• I haven't read the references, but I guess some invariance of the other metrics may be important. All these metrics induce the same topology, but they may not be equivalent (the Euclidean metric is not complete, the others may be), and the group of isometries differs (it's larger for invariant metrics). Dec 11 '17 at 22:41