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 ${\mathbb{R}}^d$, we obtain a $O(d)$-approximation with at most $n2^{O(d \log k)}$simplicesofdimension$k$ or lower. In conjunction with dimension reduction techniques, our approach yields a $O($\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}$ (n))$-approximationofsize$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 approximate filtration: we construct a point set for which every $(1+ϵ)$-approximation of the \v{C}ech filtration has to contain $n^{\mathrm{\Omega }(\log \log n)}$ features, provided that $\epsilon <\frac{1}{\log^{1+c} n}$for$c\in(0,1)$.