Applied Topology Seminar

A central question in topological data analysis is whether the sublevel-set persistent homology of a function from a sufficiently regular metric space can be recovered from a finite point sample. A natural approach is to equip the Vietoris–Rips complex of the sample, at a fixed scale, with an appropriate filtration function and to compute persistent homology of the resulting filtered complex.
 

Despite its appeal, this approach has so far lacked theoretical guarantees. Existing results instead rely on image persistence, computing the image of transition morphisms between Rips homology at two different scales. By contrast, Latschev’s theorem in metric inference shows that, under suitable regularity and sampling assumptions, the Vietoris–Rips complex of the sample at a single scale is already homotopy equivalent to the underlying space.
 

In this talk, I will explain how tools from persistent homotopy theory yield a persistent version of Latschev’s theorem, which in particular resolves this classical question of estimating persistent homology at the level of persistent homotopy types.