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.
翻译:地形数据分析(TDA)提供了新颖的方法,使我们能够分析数据集的几何形状和地形结构。作为一个重要应用,TDA可用于数据可视化和减少尺寸。我们遵循循环协调代表框架,这使我们能够使用持久性共振法对横滨上的高维数据集进行尺寸缩小和可视化。我们在本文件中建议了一种方法来调整循环协调框架,以考虑到圆形坐标在变化点和高维应用中的粗糙性。我们使用一种普遍惩罚功能,而不是传统的圆形协调算法中的$L ⁇ 2}罚款。我们提供模拟试验和实际数据分析,以支持我们的主张,即圆形坐标与普遍刑罚相协调,将探测不同取样方案下高维数据集的变化,同时保留表层结构。