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:

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

Specifically in the case where the first inequality is true simultaneously for every $k$.

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:

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

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:

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

Specifically in the case where the first inequality is true simultaneously for every $k$.

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:

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

Specifically in the case where the first inequality is true simultaneously for every $k$.