Title: Persistence Flamelets: multiscale Persistent Homology for kernel density exploration,
Abstract: 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, 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 Flamelets, a multiscale version of one of the most
popular tool in TDA, the Persistence Landscape. We examine its theoretical
properties and we show how it could be used to gain insights on KDEs bandwidth
parameter.