Persistence Curves: A canonical framework for summarizing persistence diagrams

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space is complicated. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general framework of vectorizing diagrams that we call the \textit{Persistence Curves} (PCs). We show that some well-known summaries, such as Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it. Second, we provide a theoretical foundation for the stability analysis of PCs. In addition, we propose several new summaries based on PC framework and investigate their stability. Finally, we demonstrate the practical uses of PCs on the texture classification on four public available texture datasets. We show the result of our proposed PCs outperforms several existing TDA methods.