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

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。脑机接口（BCI）是一种基于计算机的系统，它在人脑和外部设备之间建立通信通路。脑机接口（BCI）获取大脑信号，并将其转换为外部设备执行用户预期动作的命令。脑机接口在帮助患有严重运动障碍的人、检测和诊断健康问题以及为游戏等应用程序提供新的接口方面显示出了巨大的潜力。脑机接口数据的处理方法有很多，但大多是空间和时间的，如多通道脑电图和功能磁共振成像。不幸的是，由于对伪像、噪声和异常值的敏感性，现有的技术仍然存在鲁棒性和可靠性较低的问题，并且需要长时间的校准。这些挑战将通过开发一个新的框架来解决，该框架将与BCI数据的空间和时间模式相关的协方差矩阵作为正半定（PSD）矩阵流形上的元素进行有效和稳健的建模。在这种新框架下，脑机接口的处理和校准时间将大大减少，系统对小扰动的鲁棒性将增强，有可能极大地造福于患有严重运动障碍的人。这种基于流形的框架可以广泛应用于其他学科，包括生物学、农业、神经科学和计算机视觉。这项研究结合了数学、统计学和计算方法的思想，从而吸引了不同背景的学生，并帮助扩大了代表性不足群体对STEM的参与。研究结果将被整合到本科生和研究生的数据科学课程中；本研究的数学基础和计算实现将通过出版物、会议演示和开源代码传播。所使用的新框架是PSD矩阵的流形，配备了一个新的度量，称为BW （BW）度量。该公式的优点是，计算由BW度量引起的距离的速度是通常黎曼距离的两倍，因此更有效，同时对小扰动具有鲁棒性。以下的研究主题将探讨。首先研究了PSD矩阵流形上的BW距离及其诱导几何的数学性质。将开发三种算法来计算PSD矩阵集合的质心（在BW距离下）。然后将在PSD矩阵的流形上引入一类类高斯分布，并通过极大似然估计研究统计推断理论。为了将PSD矩阵分类为与BCI用户的不同操作相关的不同组，将开发高斯混合模型，并在流形切线空间上的核函数的帮助下使用非参数方法。最后，提出了两种在流形上生成合成PSD矩阵的方法，以缩短BCI的校准时间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.