  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.
