The persistence landscape and some of its properties

TL;DR

This paper explores the properties of persistence landscapes, a tool in topological data analysis, highlighting their stability, invertibility, and potential for kernel methods in statistical learning.

Contribution

It introduces a weighted version of persistence landscapes, defines related kernels, and investigates their mathematical properties and reconstructive capabilities.

Findings

Persistence landscapes are stable and invertible.

A weighted version and kernels for persistence landscapes are proposed.

Persistence landscapes can be reconstructed from their averages in many cases.

Abstract

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.