Title: The Persistent Homotopy Type Distance,
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.