Polynomial-Sized Topological Approximations Using the Permutahedron

Abstract

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(dlogk) per scale and n2 O(dlogd) 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 the Čech filtration has to contain n Ω(loglogn) features, provided that ε<1log1+cn for c∈ (0 , 1).