On Vietoris–Rips complexes of ellipses

Abstract

Abstract. For X a metric space and r > 0 a scale parameter, the Vietoris-Rips simplicial complex VR<(X; r) (resp. VR ≤ (X; r)) has X as its vertex set, and a finite subset σ ⊆ X as a simplex whenever the diameter of σ is less than r (resp. at most r). Though Vietoris-Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris-Rips complexes of ellipses Y = {(x, y) ∈ R 2 | (x/a) 2 + y 2 = 1} of small eccentricity, meaning 1 < a ≤ √ 2. Indeed, we show there are constants r 1 < r 2 such that for all r 1 < r < r 2 , we have VR<(Y ; r) S 2 and VR ≤ (Y ; r) 5 S 2 , though only one of the two-spheres in VR ≤ (Y ; r) is persistent. Furthermore, we show that for any scale parameter r 1 < r < r 2 , there are arbitrarily dense subsets of the ellipse such that the Vietoris-Rips complex of the subset is not homotopy equivalent to the Vietoris-Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

Persistent Homology with Selective Rips complexes detects geodesic circles

“…Conversely, how to interpret elements of PH in terms of geometric properties? Some of the few settings in which such an interplay has been theoretically explained contain 1-dimensional PH of metric graphs [9], 1-dimensional PH and persistent fundamental group of geodesic spaces [14,15], the complete persistence of S 1 [1], and parts of PH of ellipses [3] and regular polygons [5].…”

Contractions in persistence and metric graphs

“…Inclusions of persistence modules do not induce inclusions of barcodes or persistence diagrams. For example, over R, F [2,3] can be included into F [1,3] yet the persistence diagrams are disjoint. Inclusions of persistence diagrams have been shown to only prolong the "embedding bars" to the left (and not to the right) in case of pointwise finite-dimensional persistence modules [5].…”

“…The entire homotopy type of a Rips filtration is essentially only known in one non-trivial case: S 1 [1]. The methods of [1] can be used to extract some further results on ellipses [2] and regular polygons [3]. The entire 1-dimensional persistent homology (and fundamental group) of geodesic spaces has been completely classified in [16,17].…”

Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stability

“…, y k , y k+1 = • and choose its filling α. Let L denote a based p-sample of α, so that no point of L lies on the midpoint of any of the filled geodesic segment between consecutive vertices of L S , i.e., no vertex of L lies on the midpoint of α i , where the latter is the geodesic segment between y i and y i+1 defining α. Map L by ν π1,S p so that each vertex on α i is mapped to the closer endpoint of α i , which means either y i or y i+1 , according to Definition 3.1 (2). This implies that L is mapped to L S with repetitions of points, for example • = y 0 , y 0 , y 1 , y 1 , .…”

“…However, it is clear that such results are not always obtainable. Results of [1,2] imply that finiteness results can not hold for higher-dimensional persistences via closed filtrations. It is also apparent that compactness is required for our finiteness results.…”