I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\log(Y^{-1/2}XY^{-1/2}\|$, where the norm is the Frobenius norm.

I understand that the two metrics are closely related, and that the LE metric is something like a linearization of the Riemannian metric, but I haven't been able to find a reference that makes this precise.  Any comments or suggestions are appreciated.

For some references, [[1]] is a good intro to different metrics for SPD matrices, and [[2]] gives some more details on the context I'm working in.


  [1]: https://arxiv.org/pdf/0910.1656.pdf
  [2]: http://users.cecs.anu.edu.au/~hongdong/CVPR2013_SPD.pdf