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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?

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  • $\begingroup$ can you provide references to what you are referring as 'There are papers'? $\endgroup$
    – Thomas
    Dec 11 '17 at 19:27
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    $\begingroup$ I had put the references in the first sentence as 1, 2. I made them more noticeable now $\endgroup$
    – Damian
    Dec 11 '17 at 20:29
  • $\begingroup$ 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). $\endgroup$ Dec 11 '17 at 22:41
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It all depends on the application. Those metrics allow you to:

  1. Confine your values to the SPD manifold, an open set, while avoiding a hard boundary, which is very useful in non-linear optimization problems.
  2. Have more meaningful distance comparisons between SPD matrices. Properties such as eccentricity would diverge as you approach the boundary in Euclidean space.
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    $\begingroup$ What makes the SPD cone a manifold? I guess my issue is my very limited understanding of what a manifold is. I always pictured it as a "curved sheet" like the shell of a sphere but not the sphere itself; therefore, the idea of calling the PSD cone a manifold was confusing. I understand that these norms allows you to change the way you compare SPD matrices which can be very useful. $\endgroup$
    – Damian
    Sep 26 '18 at 17:39
  • $\begingroup$ As an example, you could create a smooth 1-1 mapping from the interior points of the unit sphere to R^3 by stretching them to infinity. Thus they are both examples of the same topological manifold. $\endgroup$ Sep 26 '18 at 20:31

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