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.

中文翻译：

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切线空间滤波器，用于可解释和有效的黎曼分类。

目的基于黎曼几何的方法已被证明是用于人机界面（BCI）解码的良好模型。但是，这些方法遭受维度的困扰，无法部署在高密度的在线BCI系统中。另外，由于缺乏黎曼方法的可解释性，因此存在人工产物驱动分类性能的可能性，这在人工控制至关重要的领域（例如患者群体的神经反馈和BCI）中存在问题。方法我们严格证明了切线空间上的任何线性函数与相应的导出空间滤波器之间的精确等价性。在此基础上，我们进一步提出了一套无需密集优化步骤的黎曼方法降维解决方案。使用开放式BCI分析框架，针对经典的常见空间模式和切线空间分类，对建议的管道进行了验证，该框架总共包含七个数据集和200个主题。最后，通过可视化相应的空间模式来验证我们框架的鲁棒性。主要结果与传统的切线空间分类相比，建议的空间滤波方法具有竞争性，有时甚至稍好一些，同时在测试阶段可将时间成本降低多达97％。重要的是，所提出的空间滤波方法的性能仅需使用4至6个滤波分量即可实现收敛，而与通道的数量无关，这也可以通过可视化的空间模式进行交叉验证。这些结果揭示了每个记录会话中潜在的神经元来源的可能性。意义我们的工作促进了对基于黎曼几何的BCI分类的理论理解，并允许更有效的分类以及从基于黎曼方法的分类器中删除伪影源。