# The Persistent Homotopy Type Distance

## TL;DR

The paper introduces the persistent homotopy type distance dHT, a new metric for comparing functions on homotopy equivalent spaces, which bounds the bottleneck distance of persistence diagrams and extends existing distances.

## Contribution

It defines the novel persistent homotopy type distance dHT, connecting homotopy theory with persistent homology, and demonstrates its properties and relations to existing metrics.

## Key findings

- dHT bounds the bottleneck distance between persistence diagrams
- dHT extends the L-infinity and natural pseudo-distances
- dHT can be interpreted via interleavings in persistence theory

## Abstract

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the L-infty distance and the natural pseudo-distance dNP. From a different standpoint, we prove that dHT extends the L-infty distance and dNP in two ways. First, we show that, appropriately restricting the category of objects to which dHT applies, it can be made to coincide with the other two distances. Finally, we show that dHT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.