Methods based on Riemannian geometryhave proven themselves to be good models for decoding in brain-computer interfacing (BCI). However, these methods suffer fromthe curse of dimensionality and are not possible to deploy inhigh-density online BCI systems. In addition, the lack of inter-pretability of Riemannian methods leaves open the possibility thatartifacts drive classification performance, which is problematic inthe areas where artifactual control is crucial, e.g., neurofeedbackand BCIs in patient populations.Approach: We rigorously provedthe exact equivalence between any linear function on the tangentspace and corresponding derived spatial filters. Upon which,we further proposed a set of dimension reduction solutions forRiemannian methods without intensive optimization steps. Theproposed pipelines are validated against classic common spatialpatterns and tangent space classification using an open-accessBCI analysis framework, which contains over seven datasets and200 subjects in total. At last, the robustness of our framework isverified via visualizing the corresponding spatial patterns.Mainresults: Proposed spatial filtering methods possess competitive,sometimes even slightly better, performances comparing to classictangent space classification while reducing the time cost upto 97% in the testing stage. Importantly, the performances ofproposed spatial filtering methods converge with using only fourto six filter components regardless of the number of channelswhich is also cross validated by the visualized spatial patterns.These results reveal the possibility of underlying neuronal sourceswithin each recording session.Significance: Our work promotesthe theoretical understanding about Riemannian geometry basedBCI classification and allows for more efficient classification aswell as the removal of artifact sources from classifiers built onRiemannian methods

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