Before revision:  Under this framework, certain  well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs.
After revision:  Under this framework, certain (PCs). We show that some well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs.