Geodesics in persistence diagram space

TL;DR

This paper investigates the structure of geodesics in the space of persistence diagrams under various metrics, showing that many geodesics are convex combinations, while others are not, depending on the metric.

Contribution

The paper characterizes when geodesics in persistence diagram space are convex combinations and constructs examples of non-convex geodesics for certain metrics.

Findings

Geodesics are convex combinations for several metrics.

Explicit examples of non-convex geodesics are provided.

The structure of geodesics depends on the choice of metric.

Abstract

It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex combination. More specifically, given two persistence diagrams and a choice of metric, one obtains a bijection realizing the distance between the diagrams, and uses this bijection to linearly interpolate from one diagram to another. We prove that for several families of metrics, every geodesic in persistence diagram space arises as such a convex combination. For certain other choices of metrics, we explicitly construct infinite families of geodesics that cannot have this form.