Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics

TL;DR

This work addresses the robustness-complexity trade-off in topological data analysis by approximating the density-sensitive subdivision-Rips bifiltration $\mathcal{SR}(X)$ with a manageable, $(1+\epsilon)$-approximation $\mathcal{NA}(X)$ for finite metric spaces of constant doubling dimension. The core approach passes through a two-stage construction: first approximate the Rips filtration $\mathcal{R}(X)$ with a compact filtration $\mathcal{A}(X)$ that uses a carefully spaced set of radii to bound the number of distinct simplicial complexes, and then apply nerve-model machinery to obtain $\mathcal{NA}(X)$, a semifiltration weakly equivalent to $\mathcal{SR}(X)$ with $k$-skeleton size $O(|X|^{k+2})$. The main theoretical contribution is proving that such an approximation exists and can be computed in time $O(|X|^{k+3})$ for fixed $k$, under the doubling-dimension assumption, with a bifiltration-size implication $O(|X|^{k+2}\log|X|)$ after converting to a bifiltration. The work leverages well-separated pair decompositions to control distinct distances and uses nerve-model results to translate a sparse cover into an efficient, interleaved representation, enabling scalable, density-sensitive persistent analysis on doubling metrics.

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 $ε>0$, we construct a $(1+ε)$-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 $O(|X|^{k+3})$.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Remark 2.1
• Definition 2.2
• Lemma 2.3: lesnickNerveModels2024, Proposition 2.10
• Theorem 2.5: lesnickNerveModels2024, Theorem 1.4
• Definition 2.6: GKL03Ass83
• Lemma 2.7: Packing Lemma eppsteinOptimalSpanners2022asmidWeakGap2009a
• Definition 2.8: callahan1995decompositiontalwar2004bypassing
• Theorem 2.9: har-peledFastConstruction2005