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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。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路径。脑-机接口获取大脑信号，并将其翻译成命令，供外部设备(S)执行用户预期的动作。BCI在帮助严重运动障碍患者、检测和诊断健康问题以及为游戏等应用程序提供新界面方面显示出巨大的潜力。许多方法已经被用来处理BCI数据，这些数据本质上是空间和时间的，例如，多通道脑电和功能磁共振成像。遗憾的是，由于对伪影、噪声和离群值的敏感性，现有技术仍然存在稳健性和可靠性低的问题，并且需要长时间的校准。这些挑战将通过开发一种新的框架来解决，该框架将与BCI数据的空间和时间模式相关联的协方差矩阵建模为正半正定(PSD)矩阵流形上的元素。在这种新的框架下，脑机接口处理和校准时间将显著减少，系统对微小扰动将变得更加健壮，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，从而吸引了具有不同背景的学生，并有助于扩大代表不足的群体在STEM中的参与。这些成果将被整合到本科生和研究生的数据科学课程中；这项研究的数学基础和计算实现将通过出版物、会议演示和开放源代码进行传播。所使用的新框架是PSD矩阵流形的框架，该流形配备了称为Bures-Wasserstein(BW)度量的新度量。这个公式的优点是，由BW度量引起的距离的计算速度是通常的黎曼距离的两倍，因此更高效，同时对小扰动具有鲁棒性。将探讨以下研究主题。首先研究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矩阵集合的重心(在BW距离下)。然后在PSD矩阵的流形上引入一类类高斯型分布，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用户期望的不同动作相关联的不同组，将开发高斯混合模型，并借助流形切线空间上的核函数使用非参数方法。最后，将开发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NSF的法定使命，并通过使用基金会的智力优势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.