The persistent homotopy type distance

We introduce the persistent homotopy type distance d_HT to compare two real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of d_HT 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 homotopy equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that d_HT 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 max-norm distance and the natural pseudo-distance d_NP. From a different standpoint, we prove that d_HT extends the max-norm distance and d_NP in two ways. First, we show that, appropriately restricting the category of objects to which d_HT applies, it can be made to coincide with the other two distances. Finally, we show that d_HT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.