Spectral Moment Embedding: A canonical generalisation of Characteristic Response Analysis for covariance-based signal classification

Covariance matrices underpin multivariate signal classifiers in brain-computer interfaces, clinical neurophysiology, and condition monitoring. Characteristic Response Analysis (CRA) and its Positive-Orthant Characteristic Response (POCR) vector extract a sign-invariant descriptor from a covariance matrix via spherical coordinates of a weighted eigenvector sum. Despite practical utility, these representations are provably non-injective and carry implicit approximations that have not been formally characterised. Here we introduce Spectral Moment Embedding (SME), which subsumes and generalises CRA/POCR through the theory of group-invariant polynomial maps. We prove that any feature invariant to the sign-flip symmetry group on covariance eigenvectors must factor through the matrix moment sequence, that the upper-triangle projection is generically injective, and that POCR is a structurally distorted, non-injective approximation to the first-moment diagonal. Lipschitz stability and a Stone-Weierstrass density result are established. Across five publicly available datasets spanning scalp EEG, polysomnography,electrocardiography, rotating machinery vibration, and inertial measurement, SME outperforms POCR/CRA with accuracy improvements of up to 30.6 percentage points. A principled negative result is also reported: static epoch-level SME features are insufficient for motor imagery BCI decoding, where temporal covariance dynamics dominate, and SME is significantly outperformed by all baselines. Feature importance and kurtosis analyses confirm moment-order interpretations predicted by theory.