Beyond motor imagery: ERP and other paradigms
Covariance features are not just for oscillatory motor imagery. For evoked responses like the P300, the raw covariance is uninformative — the signal is a time-locked template, not a change in variance. The trick is the ERP covariance representation: prepend a class-average evoked template (often estimated with the xDAWN spatial filter) to each epoch, then take the covariance of that augmented 'super-trial.' The template's projection onto the trial reappears as off-diagonal covariance, so the same MDM / tangent-space machinery now decodes ERPs.
Geometry-aware neural networks
The obvious next question is whether deep learning can improve on a hand-built tangent-space classifier. One answer keeps the geometry inside the network: SPD neural networks stack layers that map SPD inputs to SPD outputs — a bilinear mapping layer to reduce dimension, an eigenvalue-rectification layer to keep matrices positive-definite, and a matrix-log layer that plays the role of the tangent-space projection before a final linear classifier. It is, in effect, a learnable version of the pipeline you built in guides 2–3.
Be honest about where this stands. Geometry-aware networks are elegant and sometimes win, but their gains over a well-tuned tangent-space classifier are often modest, and they are data-hungry in a field where labeled data is scarce. Their main promise is transfer and joint feature learning, not a guaranteed accuracy jump.
Open problems and honest limits
The first hard limit is dimensionality. Tangent features grow as n(n+1)/2, so a 64- or 128-channel montage produces thousands of features from a covariance that itself needs \sim n^2/2 well-conditioned numbers to estimate. Both estimation and classification hit the curse of dimensionality; you must lean on shrinkage, geodesic filtering, or channel/dimension reduction. High-density EEG and ECoG grids are where Riemannian methods strain the hardest.
The second is that re-centering only fixes a shift in the mean. Real non-stationarity also warps the shape of the class distributions and can move the classes relative to each other (label-conditional shift). When yesterday's classes rotate or spread differently today, whitening to the identity is not enough, and you are back in the open problem of full distribution alignment — an active research area with no clean, unsupervised solution.
Third, the choice of geometry itself is open. AIRM and log-Euclidean are not the only metrics: the Bures–Wasserstein metric, borrowed from optimal transport, gives a different (and sometimes better-behaved) notion of distance and mean, and principled parallel transport lets you carry tangent vectors between different reference points — essential when aligning many subjects rather than re-centering each in isolation. Which metric best matches EEG statistics is still being worked out.
Why it matters — the practical bottom line
Step back and the picture is clear. Riemannian methods turned covariance — an object every BCI already computes — into a decoder that is robust by construction, needs almost no tuning, and transfers across sessions and subjects more gracefully than anything before it. That is why they became the default strong baseline for motor imagery and won multiple international BCI competitions with pipelines you could describe in a paragraph.