Barycenter Estimation of Positive Semi-Definite Matrices with Bures-Wasserstein Distance

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.