Many applications of significant societal impact are modeled by complex dynamical system behavior, including the (life, physical and social) sciences, medicine, economics, law, urban development, international politics and global conflict. Fortunately, recent advances in sensor technology have allowed observation of these phenomena at an unprecedented scale. Unfortunately, the volume and complexity of available data present many challenges to extracting meaningful information about these systems. Low-dimensional models serve as a useful structure for understanding the information in high-dimensional signals and systems. However, this information often changes over time, and so these models can further be improved by exploiting temporal dynamics. This project is concerned with developing new methods for tracking changing low-dimensional structure in data streams and dynamical systems, particularly in settings where the observations may be missing, incomplete, corrupted, or compressed.
The first research aim in this project is to develop new and substantially improve existing techniques for tracking low-dimensional structure and, in particular, to extend tracking capabilities far beyond conventional signals to much more general data sets with intrinsic low-dimensional structure. A second research aim is to develop new tools for tracking low-dimensional structure in systems jointly with estimating the content of time-varying signals and data sets. A third research aim is concerned with higher-dimensional and more complex dynamical systems, and the goal is to develop improved methods that exploit low-dimensional structure to perform quantitative and qualitative analysis in systems that are too complex and high-dimensional for system identification. In a fourth, educational aim, accessible K-12 outreach materials are being developed for dissemination through an online digital library.
