Function-Rips complexes and their approximation power

Given an unknown R^n-valued function f on a compact metric space X, can we estimate the persistent homology of f from a finite sample on X with known pairwise distances and function values? This question was answered more than a decade ago in the case n=1, assuming f is c-Lipschitz and X is a sufficiently regular geodesic metric space, and using filtered Rips complexes with fixed scale parameters for the approximation. Here we answer the question for arbitrary n, under similar conditions on X and f, and we investigate the choice of the scale parameters. We prove the statistical convergence of the persistent homology of a pair of filtered Rips complexes to the persistent homology of f, and we provide deviation bounds showing the quasi-minimax optimality of the estimator. Under more restrictive regularity conditions on X, we also prove the convergence of the persistent homology of a single filtered Rips complex to the persistent homology of f, and in fact we show that convergence happens already at the homotopy level, thus yielding a persistent version of Latschev's theorem on the homotopy type of Rips complexes of point samples on Riemannian manifolds.

This talk is based on two related manuscripts: one with Ethan Andre, Jingyi Li and David Loiseaux (arXiv:2412.04162); the other with Lukas Waas (in preparation).