ATD: Optimization on Flag Manifolds and Geometrically Constrained Neural Networks for Threat Detection

Mathematics brings a wide variety of tools that will be adapted and tuned for the development of new algorithms capable of addressing this variability in threat data observations. The outcome of this research will be new capabilities that will aid in the detection of potential threats in data. Graduate students will be trained in the application of theoretical mathematics to support algorithmic innovations related to an area of national need. This project integrates disparate areas at the interface of geometry, topology, geometric topology, optimization, machine learning and high-performance computing, all connected by the common thread of predictive analytics. A broad geometric framework will be considered that includes the interplay between Grassmannians, flags and Schubert varieties. Further, this foundational geometric framework will be integrated into the machine learning paradigm for building models from data. This approach exploits geometric structure in data taking the manifold learning model to the realm of manifolds built from matrix manifolds. 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.