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?
It all depends on the application. Those metrics allow you to:
- Confine your values to the SPD manifold, an open set, while avoiding a hard boundary, which is very useful in non-linear optimization problems.
- Have more meaningful distance comparisons between SPD matrices. Properties such as eccentricity would diverge as you approach the boundary in Euclidean space.