Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems

Latschev's theorem provides sufficient conditions on a metric space $M$ and $\delta > 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^{\delta}(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ and $\delta > 0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex $\mathcal{R}^{\delta}(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.