ATD: Topological Data Analysis for Threat Detection

变化点检测被广泛地定义为识别观察序列中的实例的问题，在该序列中，底层数据结构发生变化。这种变化可能是各种故意攻击和其他对手的结果，也可能与系统的随机故障有关。在犯罪检测、目标跟踪、语音识别、人类活动分析、网络安全、金融和环境监测等许多应用背景下，自然会出现变点检测问题。反过来，理解隐藏数据形状及其关联的形状动态是在不确定性和敌意规避下朝着更准确、更稳健和更可靠的威胁检测和风险分析迈出的重要一步。这个项目在这个重要的方向上提出了新的方法，通过引入拓扑数据分析的概念，特别是持久同源性的概念来表征、跟踪和测试数据生成过程中的变化。在持久同源性、功能数据分析和机器学习的界面上提出的公式，为自动检测攻击、威胁和其他异常的新的数据驱动的统计和机器学习方法提供了广泛的平台。该项目的最终目标是开发新的几何和拓扑增强的方法来检测来自各种来源的大型时空数据中的威胁，并研究它们的理论性质、有限样本性能和在广泛的现实世界场景中的实用性。该项目将提供一些独特的机会来促进数学科学的跨学科研究培训，特别关注让传统上代表性不足的群体在所有教育水平上参与和扩大参与。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Change point detection is broadly defined as the problem of identifying instances within a sequence of observations, where changes in the underlying data structure occur. Such changes may be a result of various intentional attacks and other adversaries or may be associated with random failures of the system. The change point detection problem naturally arises in the context of many applications such as crime detection, target tracking, speech recognition, human activity analysis, cybersecurity, financial and environmental monitoring. In turn, understanding hidden data shape and associated shape dynamics is an important step towards more accurate, robust and reliable threat detection and risk analysis under uncertainty and adversarial evasion. One graduate student will be supported each year of this grant.This project proposes novel approaches in this important direction by introducing concepts of topological data analysis and, in particular, persistent homology to characterize, track and test for changes in the data generating process. The proposed formulation at the interface of persistent homology, functional data analysis, and machine learning, offers a broad platform for new data-driven statistical and machine learning methods for automatic detection of attacks, threats, and other anomalies. The ultimate goal of the project is to develop novel geometrically and topologically enhanced procedures for detecting threats in large spatiotemporal data from a variety of sources, and to study their theoretical properties, finite-sample performance, and utility in a broad range of real-world scenarios. The project will offer a number of unique opportunities to facilitate interdisciplinary research training in mathematical sciences, with a particular focus on involving and broadening participation of traditionally underrepresented groups at all educational levels.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.