Polynomial-Sized Topological Approximations Using The Permutahedron

TL;DR

This paper introduces a new method using the permutahedron to create topological approximations of point cloud filtrations with significantly reduced size, enabling scalable analysis of high-dimensional data.

Contribution

It presents a novel geometric approach for approximating Rips complexes with polynomial size bounds, improving scalability for topological data analysis.

Findings

Achieves $O(d)$-approximation with at most $n2^{O(d \, log \, k)}$ simplices

Provides a polylogarithmic approximation for Rips filtrations on arbitrary metric spaces

Establishes a lower bound showing large size requirements for certain filtrations

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 $\mathbb{R}^d$, 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 $O(\mathrm{polylog} (n))$-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 \v{C}ech filtration has to contain $n^{\Omega(\log\log n)}$ features, provided that $\epsilon <\frac{1}{\log^{1+c} n}$ for $c\in(0,1)$.