The Space of Persistence Diagrams on $n$ Points Coarsely Embeds 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.

Error in a proof corrected, see the note in the introduction for details

Cited by in corpus (1)

• Topological and metric properties of spaces of generalized persistence diagrams