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 Rd, we obtain a O(d)-approximation whose k-skeleton has size n2O(dlogk) per scale and n2O(dlogd) in total over all scales. 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 approximation: we construct a point set for which every (1+epsilon)-approximation of the ech filtration has to contain n(loglogn) features, provided that epsilon < 1log1+cn for c(0,1).