Revisiting Riemannian geometry-based EEG decoding through approximate joint diagonalization.

Objective. The wider adoption of Riemannian geometry in electroencephalography (EEG) processing is hindered by two factors: (a) it involves the manipulation of complex mathematical formulations and, (b) it leads to computationally demanding tasks. The main scope of this work is to simplify particular notions of Riemannian geometry and provide an efficient and comprehensible scheme for neuroscientific explorations. Approach. To overcome the aforementioned shortcomings, we exploit the concept of approximate joint diagonalization in order to reconstruct the spatial covariance matrices assuming the existence of (and identifying) a common eigenspace in which the application of Riemannian geometry is significantly simplified. Main results. The employed reconstruction process abides to physiologically plausible assumptions, reduces the computational complexity in Riemannian geometry schemes and bridges the gap between rigorous mathematical procedures and computational neuroscience. Our approach is both formally established and experimentally validated by employing real and synthetic EEG data. Significance. The implications of the introduced reconstruction process are highlighted by reformulating and re-introducing two signal processing methodologies, namely the 'Symmetric Positive Definite (SPD) Matrix Quantization' and the 'Coding over SPD Atoms'. The presented approach paves the way for robust and efficient neuroscientific explorations that exploit Riemannian geometry schemes.