Medians of populations of persistence diagrams

TL;DR

This paper investigates the median of a set of persistence diagrams in Topological Data Analysis, providing a characterization of the median and comparing its properties to the mean, addressing statistical analysis challenges in complex diagram spaces.

Contribution

It introduces a new approach to defining and characterizing the median of persistence diagrams, advancing statistical tools in Topological Data Analysis.

Findings

Characterized local minima of the median's cost function

Compared properties of median and mean of diagrams

Provided insights into statistical analysis of diagram populations

Abstract

Persistence diagrams are common objects in the field of Topological Data Analysis. They are topological summaries that capture both topological and geometric structure within data. Recently there has been a surge of interest in developing tools to statistically analyse populations of persistence diagrams, a process hampered by the complicated geometry of the space of persistence diagrams. In this paper we study the median of a set of diagrams, defined as the minimizer of an appropriate cost function analogous to the sum of distances used for samples of real numbers. We then characterize the local minima of this cost function and in doing so characterize the median. We also do some comparative analysis of the properties of the median and the mean.