Affine-invariant Riemannian metric (AIRM)
The affine-invariant Riemannian metric (AIRM) is the canonical metric on the SPD manifold. Its geodesic distance between P and Q is the Frobenius norm of the matrix logarithm of P^{-1/2} Q P^{-1/2}, that is delta(P,Q) = || log(P^{-1/2} Q P^{-1/2}) ||_F, equal to the square root of the sum of squared logarithms of the generalized eigenvalues of the pair (P, Q).
Its defining property is affine (congruence) invariance: for any invertible matrix A, delta(A P A^T, A Q A^T) = delta(P,Q). The geometry is therefore unchanged by any invertible linear transform of the sensors, such as re-referencing, linear mixing, or a linear change of the volume-conduction and lead-field, which is precisely why Riemannian decoders resist the electrode-placement and session changes that wreck Euclidean methods. It is also invariant under inversion, delta(P,Q) = delta(P^{-1}, Q^{-1}).
Left- and right-multiply every trial covariance by an arbitrary invertible mixing A, as if the electrodes were relinked: AIRM distances between trials are literally unchanged, so an MDM decoder built on them is unaffected too.
Affine invariance: a linear sensor remix leaves the geometry intact.
AIRM is theoretically ideal but computationally heavier (repeated matrix square roots and eigendecompositions) and can be sensitive to poorly estimated small eigenvalues; log-Euclidean and Bures-Wasserstein are cheaper approximations or alternatives.