# The Space of Persistence Diagrams on $n$ Points Coarsely Embeds into Hilbert Space

## TL;DR

This paper investigates the geometric properties of the space of persistence diagrams, showing it coarsely embeds into Hilbert space when the number of points is fixed, but not when unbounded, with implications for topological data analysis.

## Contribution

It proves that the space of persistence diagrams with fixed points coarsely embeds into Hilbert space, and characterizes the asymptotic dimension for unbounded cases.

## Key findings

- Space of persistence diagrams on n points coarsely embeds into Hilbert space.
- Unbounded persistence diagram spaces do not have finite asymptotic dimension.
- For the bottleneck distance, the space does not coarsely embed into Hilbert space.

## Abstract

We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.