Generalized Penalty for Circular Coordinate Representation

TL;DR

This paper introduces a generalized penalty function for circular coordinate representation in Topological Data Analysis, improving change-point detection and topological structure preservation in high-dimensional data visualization.

Contribution

It proposes a novel generalized penalty approach for circular coordinates, enhancing change detection and topological fidelity in high-dimensional TDA applications.

Findings

Enhanced change-point detection in high-dimensional data

Preservation of topological structures during dimension reduction

Effective performance demonstrated through simulations and real data

Abstract

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.