Optimal rates of convergence for persistence diagrams in Topological Data Analysis
This work provides foundational statistical theory for topological data analysis, which is incremental but essential for applying persistent homology in broader statistical contexts.
The paper tackles the problem of establishing statistical convergence properties for persistence diagrams in topological data analysis, showing that they can serve as statistics with convergence guarantees in general metric spaces.
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.