Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics

TL;DR

This paper introduces a sparse approximation method for the subdivision-Rips bifiltration in metric spaces with bounded doubling dimension, achieving efficient computation while maintaining robustness to outliers.

Contribution

It presents a novel sparse approximation of the subdivision-Rips bifiltration with controlled size and computational complexity for doubling metric spaces.

Findings

Approximation preserves homotopy interleaving within (1+ε)

The k-skeleton size is polynomial in |X| for fixed k

Computational complexity is polynomial in |X| for fixed k

Abstract

The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers. Sheehy's subdivision-Rips bifiltration $\mathcal{SR}(-)$ is a density-sensitive refinement that is robust to outliers in a strong sense, but whose 0-skeleton has exponential size. For $X$ a finite metric space of constant doubling dimension and fixed $\epsilon>0$, we construct a $(1+\epsilon)$-homotopy interleaving approximation of $\mathcal{SR}(X)$ whose $k$-skeleton has size $O(|X|^{k+2})$. For $k\geq 1$ constant, the $k$-skeleton can be computed in time $O(|X|^{k+3})$.