# Upper bounds on the Gromov–Hausdorff distance using persistent homology

In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck distance between persistence diagrams in terms of the Gromov-Hausdorff distance between point clouds:

Theorem: Let $$P, Q \subset \mathbb{R}^d$$ be finite point clouds and let $$\mathrm{Filt}(⋅)$$ be any of the Čech filtration, Vietoris-Rips filtration, or alpha shape filtration. Then for any non-negative integer $$k$$,

$$d_B\left(\operatorname{dgm}\left(H_k(\operatorname{Filt}(P))\right), \operatorname{dgm}\left(H_k(\operatorname{Filt}(Q))\right)\right) \leq d_{\mathrm{GH}}(P, Q).$$

Where $$d_B$$ is the bottleneck distance and $$d_{\mathrm{GH}}$$ is the Gromov-Hausdorff distance.

I'm wondering if there exists any function $$g$$ such that the following reverse inequality also holds:

$$d_{\mathrm{GH}}(P, Q) \leq g\left(d_B\left(\operatorname{dgm}\left(H_k(\operatorname{Filt}(P))\right), \operatorname{dgm}\left(H_k(\operatorname{Filt}(Q))\right)\right)\right).$$

For general $$P$$ and $$Q$$, you won't find such a function $$g$$. For example, if you remove the assumption that $$P$$ and $$Q$$ are finite, then you could let $$P$$ be $$\mathbb{R}$$ and you could let $$Q$$ be a point. Then $$d_\mathrm{GH}(P,Q)=\infty$$ but the bottleneck distance between the persistence diagrams of $$P$$ and $$Q$$ is zero.
If you insist on $$P$$ and $$Q$$ being finite, then you could let $$P_n=\{0,1,2,3,\ldots,n\}\subseteq\mathbb{R}$$ and you could let $$Q$$ be a point. Then $$d_\mathrm{GH}(P_n,Q)\to\infty$$ as $$n\to \infty$$ but the bottleneck distance between the persistence diagrams of $$P$$ and $$Q$$ is zero for homology of dimensions $$k\ge 1$$, and the bottleneck distance between the persistence diagrams remains bounded below some fixed small constant (that does not depend on $$n$$) for homology of dimension $$k=0$$.