Geometry and Statistics on Spaces of Dynamical Systems for Pattern Recognition in High-Dimensional Time Series

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The objective of this project is to develop a mathematical framework for efficiently comparing dynamical systems identified from high-dimensional time-series data as well as algorithms for clustering, classification, and statistical analysis of such data. Dynamical systems are widely used for the analysis, verification, and control of physical, mechanical, thermal, chemical and biological processes. However, there are many emerging applications in which one also needs to "compare the dynamics of two processes". In computer vision, for example, one can use dynamical models to describe kinematic and video data of human motion. While different people move differently, the dynamical models of two people performing the same task (e.g., walking) should be "closer" to each other than the models of two people performing different tasks (e.g., walking vs running). Framework will be developed for spaces of linear dynamical systems whose quotient structure is defined by the action of a group on a smooth manifold. A family of efficiently computable distances in the ambient space will be used to define a family of "group-action-induced distances" in the quotient space. Such distances will be used to develop methods for performing classification, clustering and statistical analysis on spaces of dynamical systems. These methods will be evaluated on kinematic and video data of human activities.The development of methods for comparing dynamical models can impact both basic science and the society at large. In control theory, such methods can impact system identification and robust control. In computer vision, such techniques can be used to discriminate human and crowd activities in video data, which is relevant to many applications in surveillance, security, traffic monitoring, sports coverage/broadcast, human-computer interaction, etc. This project will train engineers and scientist in multidisciplinary research that will need concepts from differential geometry, machine learning, and computer vision. As such, it can potentially impact many other related fields. This project will also impact many diversity outreach activities, including ongoing REU programs, the Women in Science and Engineering (WISE) program and summer camps for K-12 outreach. Datasets and code will be made publicly accessible for research and educational purposes.