PERCEPT: A New Online Change-Point Detection Method using Topological Data Analysis

Topological data analysis (TDA) provides a set of data analysis tools for extracting embedded topological structures from complex high-dimensional datasets. In recent years, TDA has been a rapidly growing field which has found success in a wide range of applications, including signal processing, neuroscience and network analysis. In these applications, the online detection of changes is of crucial importance, but this can be highly challenging since such changes often occur in low-dimensional embeddings within high-dimensional data streams. We thus propose a new method, called PERsistence diagram-based ChangE-PoinT detection (PERCEPT), which leverages the learned topological structure from TDA to sequentially detect changes. PERCEPT follows two key steps: it first learns the embedded topology as a point cloud via persistence diagrams, then applies a nonparametric monitoring approach for detecting changes in the resulting point cloud distributions. This yields a nonparametric, topology-aware framework which can efficiently detect online geometric changes. We investigate the effectiveness of PERCEPT over existing methods in a suite of numerical experiments where the data streams have an embedded topological structure. We then demonstrate the usefulness of PERCEPT in two applications on solar flare monitoring and human gesture detection.

拓扑数据分析（Topological Data Analysis，TDA）提供了一套数据分析工具，用于从复杂的高维数据集中提取内嵌拓扑结构。近年来，TDA已成为快速发展的研究领域，并在信号处理、神经科学与网络分析等诸多应用领域中获得了成功应用。在这类应用中，在线变化检测是一项至关重要的任务，但该任务极具挑战性——因为此类变化往往发生在高维数据流中的低维嵌入空间内。为此我们提出了一种新方法：基于持久图的变化点检测（PERsistence diagram-based ChangE-PoinT detection，简称PERCEPT），该方法借助TDA学习得到的拓扑结构实现序列式变化检测。PERCEPT包含两个核心步骤：首先通过持久图将内嵌拓扑以点云的形式进行学习，随后采用非参数监控方法对生成的点云分布开展变化检测。该方法构建了一个非参数化、具备拓扑感知能力的框架，可高效实现在线几何变化检测。我们在一系列数据流内嵌有拓扑结构的数值实验中，对比验证了PERCEPT相较于现有方法的有效性。随后我们在太阳耀斑监测与人体手势检测两个应用场景中，展示了PERCEPT的实用价值。