CRII:CIF: Towards A Manifold-based Framework for the Brain-Computer Interface

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).A brain-computer interface (BCI) is a computer-based system that builds communication pathways between the human brain and external devices. A BCI acquires brain signals and translates them into commands for the external device(s) to perform actions intended by the user. BCIs have shown great potential for helping people with severe motor impairments, detecting and diagnosing health issues, and providing new interfaces for applications such as gaming. Numerous methods have been used to process BCI data, which are mostly spatial and temporal in nature, e.g., multi-channel electroencephalography and functional magnetic resonance imaging. Unfortunately, existing techniques still suffer from low robustness and low reliability due to sensitivity to artifacts, noise and outliers, and require lengthy calibration. These challenges will be addressed by developing a novel framework which efficiently and robustly models the covariance matrices associated with the spatial and temporal patterns of BCI data as elements on the manifold of positive semi-definite (PSD) matrices. Under this novel framework, BCI processing and calibration time will be significantly reduced, and the system will become more robust to small perturbations, with the potential to greatly benefit people suffering from severe motor impairments. This manifold-based framework can be broadly applied to other disciplines, including biology, agriculture, neuroscience, and computer vision. The research combines ideas from mathematics, statistics, and computational methods, thereby attracting students with diverse backgrounds and helping broaden the participation of underrepresented groups in STEM. The results will be integrated into both undergraduate and graduate Data Science courses; the mathematical foundations and computational implementation of this research will be disseminated through publications, conference presentations, and open-source code. The novel framework being used is that of the manifold of PSD matrices equipped with a new metric known as the Bures-Wasserstein (BW) metric. This formulation has the advantage that the computation of the distance induced by the BW metric is twice as fast as for the usual Riemannian distance, hence more efficient, while being robust to small perturbations. The following research themes will be explored. First the mathematical properties of the BW distance and its induced geometry on the manifold of PSD matrices will be studied. Three algorithms will be developed for computing the barycenter (under the BW distance) of a collection of PSD matrices. A class of Gaussian-like distributions will then be introduced on the manifold of PSD matrices, and a theory of statistical inference will be investigated through maximum likelihood estimates. To classify PSD matrices into the distinct groups associated with the different actions intended by the BCI user, Gaussian mixture models will be developed, and non-parametric approaches used with the help of kernel functions on the tangent space of the manifold. Finally, two methods for generating synthetic PSD matrices on the manifold will be developed to shorten the calibration of the BCI.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.