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.
 A: Here is a graphical description of the relation between the Riemannian and log-Euclidean metrics [ source, page 40]

The "linearization" referred to in the OP is a linearization around the identity matrix:

A: It is not so simple a relationship. Linearization suggests that log-Euclidean metric should live on some sort of tangent space to the cone of SPD matrices, but both metrics live on the cone itself instead. If $X$ and $Y$ commute then $d_{LE}(X, Y)=d_{R}(X, Y)$, and, in particular, they coincide on the tangent space to $I$. The difference is that $d_{R}$ is fully invariant under the action of $GL_n$ on the cone by conjugation $X\mapsto LXL^T$, it is the quotient metric on the cone taken as the Rimannian symmetric space $GL_n/O_n$, with $ds^2 = \text{tr}\bigl((g^{-1}dg)^Tg^{-1}dg\bigr)$ metric on $GL_n$. On the other hand, $d_{LE}$ is only invariant under the action of similarity matrices (i.e. when $L$ is a scaled orthogonal transformation). One could say that $d_{LE}$ is a poor man's $d_{R}$ with various computations (like Fréchet mean) being less costly to perform, and more stable to numerical errors. It also produces positive definite Gaussian kernels, unlike $d_{R}$.
Comparisons between properties of the two metrics are made in Log-Euclidean metrics for fast and simple calculus on diffusion tensors by Arsigny et al. In particular, determinants of Fréchet means according to both are the geometric means of the determinants (which means that geodesic interpolation does not lead to "swelling" of interpolating matrices), traces of log-Euclidean Fréchet means are always larger than of invariant ones, and the former are generally more anisotropic (the ratio of the maximal and the minimal eigenvalues is larger). Moreover, the corresponding distances are not globally uniformly equivalent even in 2D, see Equivalent metrics on symmetric positive definite matrices.
