Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Abstract

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in $$\mathbb {R}^d$$ R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the $$L_\infty $$ L ∞ -norm, which extends to a $$2d^{0.25}$$ 2 d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of $$n2^{O(d\log k +d)}$$ n 2 O ( d log k + d ) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only $$n2^{O(d)}$$ n 2 O ( d ) (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

Improved Approximate Rips Filtrations with Shifted Integer Lattices and Cubical Complexes

“…An earlier version of this paper appeared at the 25th European Symposium on Algorithms [9]. In that version, we achieved a 3 √ 2-approximation of the L ∞ Rips filtration and correspondingly, a 3 √ 2d 0.25 -approximation of the L 2 case.…”

“…In that version, we achieved a 3 √ 2-approximation of the L ∞ Rips filtration and correspondingly, a 3 √ 2d 0.25 -approximation of the L 2 case. In this version, we improve the weak interleaving of [9] to a strong interleaving to get improved approximation factors. We expand upon the details of scale balancing, among other proofs that were missing from the conference version.…”

Improved Topological Approximations by Digitization

“…The improvement in size stems from the fact that at most (d + 1)-permutahedra can intersect in d dimensions which upper bounds the number of simplices in the nerve. The approximation quality has been improved to O( 4 √ d) in subsequent work, with size n2 O(d log k) [14]. In [15], it is also shown that, for any ε < 1/ log 1+c n with c ∈ (0, 1), any ε-approximate filtration must have a size of at least n Ω(log log n) .…”

“…, improving all previous results for the Euclidean case. We achieve this approximation based on the techniques devised in [14,15]. On a fixed scale α, we tile the space with a cubical grid, carefully select a subset of them and take their nerve as our approximation complex.…”

Homology and cohomology of cubical sets with coefficients in systems of objects

“…This problem is solved using persistent homology [12], [19], and simplicial homology is traditionally used. To solve similar problems, when calculating persistent homology in a metric space, [14] used cubical homology. Calculation algorithms are more efficient than algorithms based on triangulation methods, and they can naturally be used for sets of points in an n-dimensional space with the distance between points calculated using the L ∞ norm.…”