圆坐标表示的广义罚函数
Generalized Penalty for Circular Coordinate Representation
摘要：
拓扑数据分析（TDA）提供了新的方法，使我们能够分析数据集的几何形状和拓扑结构。TDA作为一个重要的应用，可以用于数据可视化和降维。我们遵循的框架，圆坐标表示，这使我们能够执行降维和可视化的高维数据集上的环面使用持久上同调。在本文中，我们提出了一种方法来适应圆坐标框架，以考虑到圆坐标在变点和高维应用的粗糙度。我们使用一个广义罚函数代替传统圆坐标算法中的$L_{2}$罚函数。我们提供了模拟实验和真实数据分析来支持我们的主张，即具有广义惩罚的圆坐标将在不同采样方案下检测高维数据集的变化，同时保持拓扑结构。
英文摘要：
Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high-dimensional applications. We use a generalized penalty function instead of an $L_{2}$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will detect the change in high-dimensional datasets under different sampling schemes while preserving the topological structures.
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