Generating Cofaces in Vietoris--Rips Filtration Order

TL;DR

The paper tackles the inefficiency of generating Vietoris-Rips filtrations by post hoc sorting of simplices, introducing cofacet-first generation methods that preserve filtration order. It develops an in-order simplex-stream framework based on lexicographic-first full-diameter facets and employs sorted neighborhood lists to generate coboundaries directly in filtration order, while also leveraging apparent pairs to reduce work. Key contributions include three-case cofacet generation rules, a memory-efficient stack of enumerators for streaming simplices, and a coboundary generator that highlights apparent pairs early. The approach yields practical benefits for scalable persistent (co)homology computations and integrates with existing tools like Ripser, suggesting avenues for generalization to other filtered graphs and interleaved coboundary reduction.

Abstract

Cofaces -- simplices that contain a given simplex -- have multiple important uses in generating and using a Vietoris-Rips filtration: both in creating the coboundary matrix for computing persistent cohomology, and for generating the ordered sequence of simplices in the first place. Traditionally, most methods have generated simplices first, and then sorted them in filtration order after the generation step. In this paper, we propose fast algorithms for generating the sequence of simplices by generating cofaces of a given simplex with the same diameter, which by construction produces simplices in filtration order, and for generating additional cofaces in filtration order using sorted neighborhood lists in order to generate coboundaries directly in filtration order.

Figures (2)

• Figure 1: Time required to traverse each skeleton layer from 0 to 4 of a Vietoris-Rips complex on random points in $\mathbb{R}^d$. Mean and a non-parametric bootstrap confidence interval for the mean is displayed in the box, while each observation is a point: blue/left/round for the enumeration method used in Ripser and red/right/triangle for the method we propose here. Dimensions 0 and 1 are handled by directly writing vertex lists, or generating and sorting edges, while higher dimensions use the two different algorithms. Hence, the differences that can be seen in speeds for dimensions 0 and 1 are mainly due to a more elaborate setup for the inorder method.
• Figure 2: The number of vertices checked before a candidate for an Apparent Pair is found, compared between our method and the method used in Ripser. We compare the two methods with two ECDF plots: in each plot, the curve indicates the proportion of observations with value less than or equal to the $x$-coordinate. To the left, simultaneous plots of the number of vertices visited before finding an Apparent Pair candidate, we see that the filtration order methods find Apparent Pair candidates vastly much faster than the Ripser methods. To the right, we show the paired behavior: for the same choice of complex and simplex, we take the ratio of the number of candidates visited for our method, divided by the number for the method in Ripser. We note that the break-even point is at around 80-85%: in the vast majority of cases, our method is faster, in some cases by several orders of magnitude.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Theorem 3
• Definition 4
• Proposition 5
• Remark 6
• Proposition 7