Upper bounds on the Gromov–Hausdorff distance using persistent homology

Question

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:

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

Answer 1

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 $k=0$.