Persistent Homology with Selective Rips complexes detects geodesic circles

Abstract

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of S 1 ) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology in order to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.

Contractions in persistence and metric graphs

“…The assumptions of our main results of this paper are much easier to verify and in some cases hold more generally. Overall, persistent homology in dimensions 1, 2, and 3 is known to encode some geodesic circles and shortest 1-homology basis by [16,19,21] (and now also by results of this paper), properties of thick-thin decomposition [4] and injectivity radius [15]. On a similar note, the systole of a geodesic space is detected as the first critical scale of persistent fundamental group [16].…”

“…The entire 1-dimensional persistent homology (and fundamental group) of geodesic spaces has been completely classified in [16,17]. Paper [19] (and also [21]) contains a local version of the result of this paper: if a subset A ⊂ X has a sufficiently nice neighborhood, then parts of its persistent homology embed into persistent homology of X. The technical assumptions of these results hold for loops a and c of Figure 1, but not b.…”

Reconstruction properties of selective Rips complexes

“…Using ideas of [12] we prove that 1dimensional persistence of Rips and selective Rips complexes are isomorphic up to reparameterization. This is in sharp contrast with higher-dimensional persistence [13]. To summarize, we prove that the reconstruction properties of selective Rips complexes closely resemble those of Rips complexes.…”

“…However, detections in dimensions 2 and above require geodesic circles to have wide neighborhoods adhering to certain geometric conditions. This technical requirement has been circumvented in [13] which demonstrated that arbitrarily small geodesically convex neighborhoods suffice to detect a geodesic circle using selective Rips complexes with two parameters. In particular, selective Rips complexes with two parameters are a modification of Rips complexes designed to detect many (in some cases all [14]) geodesic circles.…”

“…Selective Rips complexes have a potential to act as a finer yet still easily computable version of Rips complexes. In [13] it was demonstrated that selective Rips complexes detect some geodesic circles that are undetected by Rips complexes. As our understanding of information encoded by Rips complexes (and the corresponding persistent homology) grows [11,12,15,13,14,8,2], we expect the variants using selective Rips complexes to encode more information of the same type.…”