Log-Euclidean metric
The log-Euclidean metric maps each SPD matrix through the matrix logarithm into the flat vector space of symmetric matrices, computes ordinary Euclidean distances and means there, and (when needed) maps back with the matrix exponential. Its geodesic distance is || log(P) - log(Q) ||_F and its mean is exp( mean_i log(P_i) ), both closed-form and fast, with no iterative eigen-heavy Karcher flow.
It preserves some good properties, avoiding the determinant swelling of naive Euclidean averaging and keeping results SPD, but it is invariant only to orthogonal (rotation) and scaling transforms, not to the full affine group, so it is less robust to sensor re-mixing than the affine-invariant metric. In practice it is an excellent, cheap approximation to AIRM when covariances are well-conditioned and concentrated; the gap widens for ill-conditioned matrices or data spread widely on the manifold.
Swap AIRM for the log-Euclidean metric in an MDM decoder: means become exp(mean of logs) and distances become norms of log differences, and the large speedup often costs only a point or two of accuracy on well-conditioned data.
Log-Euclidean: cheaper geometry, small accuracy cost.
The log-Euclidean tangent space is a single global logarithm (reference = identity), whereas AIRM tangent projection is taken at a data-dependent geometric mean; the two coincide near the identity.