Theorem 4.5, p. 19 — a multi-bijection D₀f → D₀𝓡^f_δ(L) moves points by ≤ cδ in Q^NE_α and abscissae by ≤ cδ in Q^SE_α

Let $\mathbb X$ be a Riemannian manifold, possibly non-compact and possibly with boundary, whose strong convexity radius $\varrho_c(\mathbb X)$ is positive. Let $L\subseteq\mathbb X$ be a finite point cloud and $f:\mathbb X\to\mathbb R$ a tame $c$-Lipschitz function ($c\ge0$). Let $0<\delta<\varrho_c(\mathbb X)$ and $\alpha\in\mathbb R$ be such that $L$ is a geodesic $\delta/4$-sample of $\mathbb F^\alpha=f^{-1}([\alpha,\infty))$. Then there is a multi-bijection $\gamma$ between the $0$-th persistence diagram $D_0f$ of the superlevel-set filtration of $f$ and the $0$-th persistence diagram $D_0\mathcal R^f_\delta(L)$ of the upper-star Rips filtration, such that

1. $\forall p\in D_0f\cap Q^{NE}_\alpha$, $\|p-\gamma(p)\|_\infty\le c\delta$;
2. $\forall q\in D_0\mathcal R^f_\delta(L)\cap Q^{NE}_\alpha$, $\|\gamma^{-1}(q)-q\|_\infty\le c\delta$;
3. $\forall p\in D_0f\cap Q^{SE}_\alpha$, $|p_x-\gamma(p)_x|\le c\delta$;
4. $\forall q\in D_0\mathcal R^f_\delta(L)\cap Q^{SE}_\alpha$, $|\gamma^{-1}(q)_x-q_x|\le c\delta$.

It is the stability step of the main theorem: above the level $\alpha$ the diagram computed from the sample approximates the diagram of $f$.

Formalization Note A multi-bijection is a bijection between the copies of the points of the two diagrams, the diagonal contributing countably many copies of each of its points $(x,x)$, $x\in[-\infty,+\infty]$; the assertions quantify over all copies. On the Rips side the vertex set is $L$ itself, with the ambient (geodesic) distance and $g=f|_L$.