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.
