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

这个项目的目标是开发一个数学框架，用于有效地比较从高维时间序列数据中识别的动态系统，以及对这些数据进行聚类、分类和统计分析的算法。动力系统被广泛用于物理、机械、热、化学和生物过程的分析、验证和控制。然而，在许多新兴的应用中，人们还需要“比较两个过程的动态”。例如，在计算机视觉中，人们可以使用动力学模型来描述人体运动的运动学和视频数据。虽然不同的人移动方式不同，但两个人执行相同任务(例如，行走)的动态模型应该比两个人执行不同任务(例如，行走或跑步)的模型彼此之间的距离更近。建立了线性动力系统空间的框架，其商结构由光滑流形上的群的作用来定义。环境空间中的一族可有效计算的距离将被用来定义商空间中的一族“群体行动诱导距离”。这些距离将被用来开发对动力系统空间进行分类、聚类和统计分析的方法。这些方法将根据人类活动的运动学和视频数据进行评估。动力学模型比较方法的发展将对基础科学和整个社会产生影响。在控制理论中，这种方法会影响系统辨识和鲁棒控制。在计算机视觉中，这种技术可用于区分视频数据中的人类和人群活动，这与许多应用程序有关，如监控、安全、交通监控、体育报道/广播、人机交互等。该项目将培训工程师和科学家进行多学科研究，需要微分几何、机器学习和计算机视觉的概念。因此，它可能会对许多其他相关领域产生潜在影响。该项目还将影响许多多元化外展活动，包括正在进行的REU方案、妇女参与科学和工程(WISE)方案和K-12外展夏令营。数据集和代码将公开供研究和教育使用。

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.