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(d log k) per scale and n2 O(d log d) 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 theČech filtration has to contain n (log log n) features, provided that ε < 1 log 1+c n for c ∈ (0, 1).

Coxeter Triangulations Have Good Quality

“…Theà d Coxeter triangulation was also used to approximate the Rips filtration of n points in R d in a recent work by Choudhary et al [11]. The authors arrived at an approximation scheme that achieved a significant reduction in size of the complex.…”

Improved Topological Approximations by Digitization

“…• The factor 2 O(d log d+dk) in our size bound can be further reduced to 2 O(d log d) by replacing the cubical grid by a permutahedral grid [15]. A consequence of this result is that when d = Θ(log n) and ε = 1/ log 1+c n, for some c ∈ (0, 1), our approximation with the permutahedral grid has a size of n O(log log n) matching the lower bound of [15].…”

“…We go over each pixel and inspect whether its 3 d − 1 neighbors lie in the dictionary; if so, we put an edge between the pixel and its neighbor. The k-skeleton can be obtained from the 1-skeleton by a simple combinatorial algorithm, traversing the Hasse diagram of the complex (refer to Algorithm 5.8 in [15]).…”

“…Choudhary et al [15] presented an alternative approximation scheme, where the approximation quality is only O(d), but the size of the approximation is significantly smaller at n2 O(d log k) . This result is obtained by tiling the space into permutahedra (a generalization of the hexagonal grid in R 2 ) where the diameter of the permutahedra is controlled by the scale parameter α.…”

“…The approximation quality has been improved to O( 4 √ d) in subsequent work, with size n2 O(d log k) [14]. In [15], it is also shown that, for any ε < 1/ log 1+c n with c ∈ (0, 1), any ε-approximate filtration must have a size of at least n Ω(log log n) .…”