Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R d , we obtain a O(d)-approximation whose k-skeleton has size n2 O(d log k) per scale and n2 O(d log d) in total over all scales. In conjunction with dimension reduction techniques, our approach yields a O(polylog(n))-approximation of size n O(1) for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximation: we construct a point set for which every (1 + ε)-approximation of theČech filtration has to contain n (log log n) features, provided that ε < 1 log 1+c n for c ∈ (0, 1).

Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/ √ d) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of nondegeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d 2). We conjecture that both bounds are optimal for triangulations in Euclidean space.

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.

AbstracťCech complexes are useful simplicial complexes for computing and analyzing topological features of data that lies in Euclidean space. Unfortunately, computing these complexes becomes prohibitively expensive for large-sized data sets even for medium-to-low dimensional data. We present an approximation scheme for (1 + ε)-approximating the topological information of theČech complexes for n points in R d , for ε ∈ (0, 1]. Our approximation has a total size of n 1 ε O(d) for constant dimension d, improving all the currently available (1 + ε)-approximation schemes of simplicial filtrations in Euclidean space. Perhaps counterintuitively, we arrive at our result by adding additional n 1 ε O(d) sample points to the input.We achieve a bound that is independent of the spread of the point set by pre-identifying the scales at which theČech complexes changes and sampling accordingly.

We introduce the notion of t-restricted doubling dimension of a point set in Euclidean space as the local intrinsic dimension up to scale t. In many applications information is only relevant for a fixed range of scales. We present an algorithm to construct a hierarchical net-tree up to scale t which we denote as the net-forest. We present a method based on Locality Sensitive Hashing to compute all near neighbours of points within a certain distance. Our construction of the net-forest is probabilistic, and we guarantee that with high probability, the net-forest is supplemented with the correct neighbouring information. We apply our net-forest construction scheme to create an approximate Čech complex up to a fixed scale; and its complexity depends on the local intrinsic dimension up to that scale.