ATD: Algorithms for Data Analysis on Abstract Manifolds

该研究项目涉及开发新的数学算法，用于识别数据集中的关键信息，这些信息有可能揭示对人类的生物或化学威胁。这项工作的出发点是观察到对其外观变化的威胁建模是鲁棒检测算法的基础。来自几何和拓扑的数学工具能够设计从数据集中提取信息的算法，从而增强传统的处理方法，从而实现更智能的传感器。在这个项目中训练的研究生将获得数学博士学位，而本科生将有机会在他们职业生涯的早期就国家感兴趣的话题进行指导。本研究解决了抽象流形上数据的数学建模算法的发展。与传感器捕获的数据云不同，传感器生成的点在欧几里得空间中以配置形式存在，该项目将数据编码为矩阵流形上的点。数学框架只需要三个特征：几何背景，两点之间距离的概念，以及计算一点在另一点方向上的扰动的能力。此外，还可以利用与矩阵流形上点的统计相关的信息。本研究计划旨在探索这些空间的几何形状，以进行异常检测和表征。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 research program concerns the development of new mathematical algorithms for identifying critical information in data sets that has the potential to reveal biological or chemical threats to humans. The starting point for this work is the observation that modeling a threat over variations in its appearance serves as the foundation of robust detection algorithms. Mathematical tools from geometry and topology enable the design of algorithms for extracting information from sets of data that enhance traditional processing methods, leading to smarter sensors. The graduate students trained in this program will earn doctorates in mathematics while the undergraduates will have the opportunity to be mentored on topics of national interest early in their careers. This research addresses the development of mathematical modeling algorithms for data on abstract manifolds. In contrast to data clouds captured by sensors generating points that exist as a configuration in Euclidean space, this project encodes data as points on matrix manifolds. The mathematical framework requires only three features: the geometric context, the notion of distance between two points, and the ability to compute the perturbation of one point in the direction of another. Additionally, one can exploit information related to the statistics of points on matrix manifolds. This research program aims to explore the geometry of these spaces for the purposes of anomaly detection and characterization.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.