ERP covariance representation
Riemannian methods were built for the covariances of oscillatory signals, where a trial's spatial covariance carries the discriminative power. Time-locked ERPs such as the P300 are different, because the discriminative information is in the average evoked waveform rather than the spontaneous covariance, so a special construction is used: each trial is stacked with class-average ERP templates (often after xDAWN spatial filtering) to form an augmented super-trial, whose covariance then encodes how strongly the trial resembles each template. This turns an ERP-detection problem back into a covariance-classification problem the Riemannian toolbox can handle.
This special-form covariance made Riemannian classifiers competitive on P300 spellers and was part of BCI-competition-winning ERP pipelines, giving a unified geometric treatment across motor imagery, SSVEP, and ERP paradigms. The cost is a larger covariance (channels times the number of classes plus templates) that needs more samples and shrinkage to estimate reliably.
For a P300 speller, prepend the target and non-target average responses to each trial and take the covariance of the stack: target trials show high covariance with the target template, and a Riemannian classifier reads that off directly.
Special-form covariance turns P300 detection into covariance classification.