论文标题
循环坐标表示的普遍罚款
Generalized Penalty for Circular Coordinate Representation
论文作者
论文摘要
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。作为一个重要的应用，TDA可用于数据可视化和降低尺寸。我们遵循循环坐标表示的框架，这使我们能够使用持久的共同体在圆环上对高维数据集执行尺寸降低和可视化。在本文中，我们提出了一种调整圆形坐标框架的方法，以考虑变化点和高维应用中圆形坐标的粗糙度。我们使用广义的惩罚功能，而不是传统圆形坐标算法中的$ L_ {2} $罚款。我们提供仿真实验和实际数据分析，以支持我们的主张，即具有广义惩罚的循环坐标将在不同的采样方案下检测高维数据集的变化，同时保留拓扑结构。
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.
