eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-06-10
31:1
31:16
10.4230/LIPIcs.SoCG.2016.31
article
Polynomial-Sized Topological Approximations Using the Permutahedron
Choudhary, Aruni
Kerber, Michael
Raghvendra, Sharath
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 with at most n2^{O(d log k)} simplices of dimension k or lower. 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 approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol051-socg2016/LIPIcs.SoCG.2016.31/LIPIcs.SoCG.2016.31.pdf
Persistent Homology
Topological Data Analysis
Simplicial Approximation
Permutahedron
Approximation Algorithms