ATD: Detection and Classification of Threats Using Subspace Manifold Geometry

这项提议涉及开发新的数学算法，用于检测和分类由于生物制剂或化学羽流的存在而产生的大数据集中的威胁。研究人员提出了一个几何框架，以对子空间流形上的海量数据集进行编码。利用给定类别的一组点可以表示为高维环境空间的低维子空间这一事实，可以捕获威胁中的更多可变性，从而以更高的精度表征它们。通过子空间表示数据的方法有很多种，每一种都导致了一个严格的流形概念，例如Grassmann流形和Stiefel流形。研究人员建议使用这些流形的几何形状作为抽象点或通过构造欧几里得空间中的嵌入来表示威胁中的模式。最初针对向量空间提出的检测和分类算法现在可以扩展到子空间流形上的算法。拟议的跨学科研究计划解决了对化学和生物威胁的检测和分类，这是国家安全面临的主要挑战。传递到城市环境或军事战区的威胁可能由未知物质组成，目标是检测、表征和跟踪威胁。或者，威胁可能与已知物质相关，目标是通过将实际材料或代理与已知威胁的签名库进行匹配，不仅检测实际材料或代理，而且对其进行分类。将根据国防威胁减少局提供的数据集对将要进行的基础研究进行评估。这些数据包括(但不限于)使用法布里-珀罗涉仪、频率捷变激光雷达和拉曼光谱获取的数据。这项研究将由科罗拉多州立大学数学系和计算机科学系的教职员工领导，为学生提供独特的多学科体验，并在教育中进行研究整合。

This proposal concerns the development of new mathematical algorithms for detecting and classifying threats in large data sets arising from the presence of, e.g., biological agents or chemical plumes. The investigators propose a geometric framework centered on encoding massive data sets on subspace manifolds. Exploiting the fact that a set of points of a given class can be represented as a low-dimensional subspace of a high dimensional ambient space, it is possible to capture more variability in the threats and thus characterize them with higher accuracy. There are many ways to represent data via subspaces, each leading to a rigorous notion of a manifold, e.g., the Grassmann and Stiefel manifolds. The investigators propose to use the geometry of these manifolds, either as abstract points or via constructing embeddings in Euclidean space, for representing patterns in threats. Detection and classification algorithms originally proposed for vector spaces may now be extended to algorithms over subspace manifolds. The proposed interdisciplinary research program addresses the detection and classification of chemical and biological threats, a major challenge for National Security. Threats delivered to an urban environment or military theater, are potentially comprised of unknown substances and the goal is to detect, characterize and track the threat. Alternatively, threats may be associated with known substances and the goal is to not only detect but classify the actual material or agent by matching it to a library of signatures of known threats. The basic research to be performed will be evaluated in the context of data sets made available by the Defense Threat Reduction Agency. These include (but are not limited to) data acquired using a Fabry-Perot Interferometer, Frequency Agile Lidar, and Raman Spectroscopy. The research will be led by faculty from the Departments of Mathematics and Computer Science at Colorado State University, providing the students with unique multidisciplinary experience with research integration in education.