# How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

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.