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Nerve Models of Subdivision Bifiltrations

Michael Lesnick, Kenneth McCabe

TL;DR

This work develops a poly-size, weakly equivalent simplicial model for subdivision bifiltrations, enabling scalable analysis of density-sensitive multiparameter persistence. It proves a general NF construction that bounds k-skeleton size by O(m_k) with a matching lower bound on the 0-skeleton, and introduces a sqrt{2}-approximation J(X) to the subdivision-Rips bifiltration with skeleta size O(|X|^{k+2}). The authors show the sqrt{2} bound is tight in general, while in fixed Euclidean dimensions with the \\ell_p metric they obtain exact poly-size models for p in {1,\\infty} and (1+ε)-approximations with poly-size skeleta for p in (1,\\infty). They also connect subdivision bifiltrations to intrinsic subdivision-Čech via interleavings, and discuss computation and practical implications for large-scale MPH analyses.

Abstract

We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We focus in particular on the subdivision-Rips bifiltration $\mathcal{SR}(X)$ of a metric space $X$, the only density-sensitive bifiltration on metric spaces known to satisfy a strong robustness property. Given a simplicial filtration $\mathcal{F}$ with a total of $m$ maximal simplices across all indices, we introduce a nerve-based simplicial model for its subdivision bifiltration $\mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the $0$-skeleton of any simplicial model of $\mathcal{SF}$ has size at least $m$. We give several applications: For an arbitrary metric space $X$, we introduce a $\sqrt{2}$-approximation to $\mathcal{SR}(X)$, denoted $\mathcal{J}(X)$, whose $k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best approximation bound of $\sqrt{3}$, achieved by the degree-Rips bifiltration, which implies that $\mathcal{J}(X)$ is more robust than degree-Rips. Moreover, we show that the approximation factor of $\sqrt{2}$ is tight; in particular, there exists no exact model of $\mathcal{SR}(X)$ with poly-size skeleta. On the other hand, we show that for $X$ in a fixed-dimensional Euclidean space with the $\ell_p$-metric, there exists an exact model of $\mathcal{SR}(X)$ with poly-size skeleta for $p\in \{1, \infty\}$, as well as a $(1+ε)$-approximation to $\mathcal{SR}(X)$ with poly-size skeleta for any $p \in (1, \infty)$ and fixed ${ε> 0}$.

Nerve Models of Subdivision Bifiltrations

TL;DR

This work develops a poly-size, weakly equivalent simplicial model for subdivision bifiltrations, enabling scalable analysis of density-sensitive multiparameter persistence. It proves a general NF construction that bounds k-skeleton size by O(m_k) with a matching lower bound on the 0-skeleton, and introduces a sqrt{2}-approximation J(X) to the subdivision-Rips bifiltration with skeleta size O(|X|^{k+2}). The authors show the sqrt{2} bound is tight in general, while in fixed Euclidean dimensions with the \\ell_p metric they obtain exact poly-size models for p in {1,\\infty} and (1+ε)-approximations with poly-size skeleta for p in (1,\\infty). They also connect subdivision bifiltrations to intrinsic subdivision-Čech via interleavings, and discuss computation and practical implications for large-scale MPH analyses.

Abstract

We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We focus in particular on the subdivision-Rips bifiltration of a metric space , the only density-sensitive bifiltration on metric spaces known to satisfy a strong robustness property. Given a simplicial filtration with a total of maximal simplices across all indices, we introduce a nerve-based simplicial model for its subdivision bifiltration whose -skeleton has size . We also show that the -skeleton of any simplicial model of has size at least . We give several applications: For an arbitrary metric space , we introduce a -approximation to , denoted , whose -skeleton has size . This improves on the previous best approximation bound of , achieved by the degree-Rips bifiltration, which implies that is more robust than degree-Rips. Moreover, we show that the approximation factor of is tight; in particular, there exists no exact model of with poly-size skeleta. On the other hand, we show that for in a fixed-dimensional Euclidean space with the -metric, there exists an exact model of with poly-size skeleta for , as well as a -approximation to with poly-size skeleta for any and fixed .
Paper Structure (24 sections, 27 theorems, 78 equations, 5 figures)

This paper contains 24 sections, 27 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Density-Sensitive Bifiltrations
  4. Contributions
  5. Preliminaries
  6. Filtrations
  7. Clique and Čech Filtrations
  8. Weak Equivalence
  9. Persistence Modules
  10. Presentations and Resolutions
  11. Chain Complexes and Homology of Filtrations
  12. Size of Simplicial Functors
  13. Subdivision Bifiltrations
  14. Homotopy Interleavings
  15. Functorial Nerves
  16. ...and 9 more sections

Key Result

Theorem 1.1

For any finite, non-empty metric spaces $X$ and $X'$, the homotopy interleaving distance between $\mathcal{SR}(X)$ and $\mathcal{SR}(X')$ is at most the Gromov-Prohorov distance between the uniform probability measures on $X$ and $X'$.

Figures (5)

  • Figure 1: The subdivision filtration of a $2$-simplex $\Delta$.
  • Figure 2: Construction of $\mathcal{NR}(X)$, for $X$ a set of four points in $\mathbb R^2$.
  • Figure 3: The cocktail party graph $C_3$.
  • Figure 4: The neighborhood graph $N(X_4)_{1-\delta}$.
  • Figure 5: Spherical vs polyhedral neighborhood graphs.

Theorems & Definitions (69)

  • Theorem 1.1: blumbergStability2Parameter2022
  • Definition 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Conjecture 1.9
  • Corollary 1.10
  • Definition 2.1
  • ...and 59 more