Can you explain to me why Riemann distance is better for positive semidefinite matrices (for example covariance matrices) than Euclidian distance?
Here is the riemannian distance: $$ d\left(Σ_A,Σ_B\right)=\sqrt{ \sum_i{\ln^2{λ_i (Σ_A,Σ_B)}}} $$ Where $$\lambda_i(Σ_A,Σ_B)$$ is the generalized eigeinvalue of A and B (see https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem).
I do not have, or little knowledge in geodesics but it seems to be the reason why it is better.
Any clarification would be very helpful
Thank you very much!