Persistence Flamelets: Topological Invariants for Scale Spaces

In recent years there has been noticeable interest in the study of the “shape of data”. Among the many ways a “shape” could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, or TDA for short, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelet, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show its performance as both an exploratory and inferential tool. In addition, we provide open source implementation of the objects and methods presented in the R-package pflamelet.