# Embeddings of Persistence Diagrams into Hilbert Spaces

## TL;DR

This paper investigates the geometric properties of persistence diagrams under the bottleneck distance, showing they cannot be coarsely embedded into Hilbert spaces and exploring their metric space characteristics.

## Contribution

It proves that persistence diagrams with the bottleneck distance do not admit coarse embeddings into Hilbert spaces and characterizes their metric properties.

## Key findings

- Persistence diagrams do not admit coarse embedding into Hilbert spaces.
- Any separable, bounded metric space can be isometrically embedded into the space of persistence diagrams.
- The space of persistence diagrams has infinite asymptotic dimension.

## Abstract

Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence diagrams with the bottleneck distance do not even admit a coarse embedding into a Hilbert space. As part of our proof, we show that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance. As corollaries, we obtain the generalized roundness, negative type, and asymptotic dimension of this space.