Brain-computer interface (BCI) builds a bridge between human brain and external devices by recording brain signals and translating them into commands for devices to perform the user's imagined action. The core of the BCI system is the classifier that labels the input signals as the user's imagined action. The classifiers that directly classify covariance matrices using Riemannian geometry are widely used not only in BCI domain but also in a variety of fields including neuroscience, remote sensing, biomedical imaging, etc. However, the existing Affine-Invariant Riemannian-based methods treat covariance matrices as positive definite while they are indeed positive semi-definite especially for high dimensional data. Besides, the Affine-Invariant Riemannian-based barycenter estimation algorithms become time consuming, not robust, and have convergence issues when the dimension and number of covariance matrices become large. To address these challenges, in this paper, we establish the mathematical foundation for Bures-Wasserstein distance and propose new algorithms to estimate the barycenter of positive semi-definite matrices efficiently and robustly. Both theoretical and computational aspects of Bures-Wasserstein distance and barycenter estimation algorithms are discussed. With extensive simulations, we comprehensively investigate the accuracy, efficiency, and robustness of the barycenter estimation algorithms coupled with Bures-Wasserstein distance. The results show that Bures-Wasserstein based barycenter estimation algorithms are more efficient and robust.

翻译：脑机接口通过记录大脑信号并将其转换为指令，使外部设备执行用户想象的动作，从而在人脑与外部设备之间建立桥梁。脑机接口系统的核心是分类器，用于将输入信号标注为用户想象的行动。直接利用黎曼几何对协方差矩阵进行分类的方法不仅在脑机接口领域广泛应用，还涉及神经科学、遥感、生物医学成像等多个领域。然而，现存的基于仿射不变黎曼几何方法将协方差矩阵视为正定矩阵，而实际上在高维数据场景下它们往往是半正定的。此外，当协方差矩阵的维度和数量增大时，基于仿射不变黎曼几何的重心估计算法会变得耗时、不鲁棒，并可能出现收敛问题。为解决这些挑战，本文为Bures-Wasserstein距离建立了数学基础，并提出了高效稳健的半正定矩阵重心估计算法。本文从理论和计算两个层面讨论了Bures-Wasserstein距离及重心估计算法。通过大量仿真实验，我们全面研究了结合Bures-Wasserstein距离的重心估计算法的精度、效率和鲁棒性。结果表明，基于Bures-Wasserstein的重心估计算法具有更高的效率和鲁棒性。