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 with at most n2O(dlogk) 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 \v{C}ech filtration has to contain
nΩ(loglogn) features, provided that ϵ<log1+cn1 for c∈(0,1).Comment: 24 pages, 1 figur