Persistent Cost of Lipschitz Maps

A $1$-Lipschitz map between compact metric spaces $f\text{\hspace{0.17em}⁣}:X\to Y$ induces a homomorphism of persistence modules on degree-$d$ Vietoris–Rips persistent homology. We define the persistent cost of $f$ from this induced homomorphism by quantifying the persistence carried by its kernel and cokernel modules. We prove that the persistent cost controls the interleaving distance between the degree-$d$ Vietoris–Rips persistent homology modules of $X$ and $Y$. Moreover, we obtain an explicit upper bound for the persistent cost in purely metric terms. Finally, we give a self-contained proof of the stability of the persistent cost introducing a Gromov-Hausdorff type distance for maps between compact metric spaces.