Tangent space spatial filters for interpretable and efficient Riemannian classification

Objective: Methods based on Riemannian geometry have proven themselves to be good models for decoding in brain-computer interfacing (BCI). However, these methods suffer from the curse of dimensionality and are not possible to deploy in high-density online BCI systems. In addition, the lack of interpretability of Riemannian methods leaves open the possibility that artifacts drive classification performance, which is problematic in the areas where artifactual control is crucial, e.g., neurofeedback and BCIs in patient populations. Approach: We rigorously proved the exact equivalence between any linear function on the tangent space and corresponding derived spatial filters. Upon which, we further proposed a set of dimension reduction solutions for Riemannian methods without intensive optimization steps. The proposed pipelines are validated against classic common spatial patterns and tangent space classification using an open-access BCI analysis framework, which contains over seven datasets and 200 subjects in total. At last, the robustness of our framework is verified via visualizing the corresponding spatial patterns. Main results: Proposed spatial filtering methods possess competitive, sometimes even slightly better, performances comparing to classic tangent space classification while reducing the time cost up to 97\% in the testing stage. Importantly, the performances of proposed spatial filtering methods converge with using only four to six filter components regardless of the number of channels which is also cross validated by the visualized spatial patterns. These results reveal the possibility of underlying neuronal sources within each recording session. Significance: Our work promotes the theoretical understanding about Riemannian geometry based BCI classification and allows for more efficient classification as well as the removal of artifact sources from classifiers built on Riemannian methods.