Polynomial-Sized Topological Approximations Using the Permutahedron

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, we obtain a O(d)-approximation with at most n2O(d log k) simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog(n))-approximation of size nO(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 approximate filtration: we construct a point set for which every (1 + ε)-approximation of the Čech filtration has to contain nΩ(log logn) features, provided that ε < 1 log1+c n for c ∈ (0, 1). 1998 ACM Subject Classification F.2.2 Geometric problems and computation