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It's All About Covers: Persistent Homology of Cover Refinements

António Leitão

Abstract

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth which can be exponential in the worst case, as for the Vietoris-Rips. We recast this problem at the level of covers, developing a framework in which filtrations and persistence modules can be constructed, analyzed, and compared through simple relations between covers rather than at the level of simplicial complexes. The guarantees propagate through any functor that preserves the contiguity of refinement maps, we give the example of two such functors: the Nerve and the Co-Nerve. Working at this level is drastically simpler, with stronger, more general consequences. We explore this perspective and show how it can be used to construct a robust approximation of the Vietoris-Rips filtration that is orders of magnitude smaller, while maintaining a log 3-interleaving unconditionally for any metric space. The resulting filtration restores near-linear scaling in the number of data points and enable us to efficiently capture homology at high degrees.

It's All About Covers: Persistent Homology of Cover Refinements

Abstract

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth which can be exponential in the worst case, as for the Vietoris-Rips. We recast this problem at the level of covers, developing a framework in which filtrations and persistence modules can be constructed, analyzed, and compared through simple relations between covers rather than at the level of simplicial complexes. The guarantees propagate through any functor that preserves the contiguity of refinement maps, we give the example of two such functors: the Nerve and the Co-Nerve. Working at this level is drastically simpler, with stronger, more general consequences. We explore this perspective and show how it can be used to construct a robust approximation of the Vietoris-Rips filtration that is orders of magnitude smaller, while maintaining a log 3-interleaving unconditionally for any metric space. The resulting filtration restores near-linear scaling in the number of data points and enable us to efficiently capture homology at high degrees.
Paper Structure (40 sections, 35 theorems, 94 equations, 8 figures, 1 table, 5 algorithms)

This paper contains 40 sections, 35 theorems, 94 equations, 8 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Covers and Dowker complexes.
  5. Sparse and reduced filtrations
  6. Simplicial Towers
  7. Relaxed interleavings
  8. Background
  9. Category Theory
  10. Simplicial complexes and simplicial maps
  11. Dowker Complexes
  12. Covers and Refinements
  13. Approximation of Covers
  14. The Nerve and Co-Nerve
  15. Persistent Covers
  16. ...and 25 more sections

Key Result

Proposition 2.4

Let $F,G:(\mathbb{R}_+,\leq)\to \boldsymbol{\mathbf{C}}$ be persistent objects and suppose there exists a $c$-approximation $(\varphi,\psi)$ between them. Define the reindexed functors and for $s\le t$ set $\widetilde{F}(s\le t):=F(e^s\le e^{t})$ (and similarly for $\widetilde{G}$). Then define natural transformations and the coherence relations eq:add-coh-1--eq:add-coh-2 hold with $\varepsilon

Figures (8)

  • Figure 4: Runtime comparison between Vietoris-Rips filtration and our method. (a)$H_2$ computation time for increasing sample sizes of a torus in $\mathbb{R}^3$. (b)$H_n$ computation time for $2000$-point samples of $n$-spheres. (c) Persistence diagram from the $5$-sphere run, capturing the $H_5$ non-trivial class.
  • Figure 5: Example of two contiguous refinement maps $f,g:\mathcal{V} \to \mathcal{U}$ defined as $f(V_1)=f(V_2)=U_1$, $f(V_3)=U_2$ and $g(V_1)=U_1$, $g(V_2)=g(V_3)=U_2$
  • Figure 6: The ball cover
  • Figure 7: The maximal clique cover. To help visualize the clique structure we added grey lines matching points $x,y$ whenever $d(x,y)\leq r$
  • Figure 8: A persistent clique-partition $P$. In this case it is a maximal clique partition (each partition is a maximal clique). We added all the edges whenever $d(x,y)\leq r$.
  • ...and 3 more figures

Theorems & Definitions (97)

  • Definition 2.1: Persistence Module
  • Definition 2.2: Interleaving chazal2009proximitylesnick2015theoryoudot2015persistence
  • Definition 2.3: $c$-Approximation
  • Proposition 2.4
  • Definition 2.5: Interleaving Distance
  • Lemma 2.6
  • proof
  • Definition 2.7: Contiguous Simplicial Maps munkres2018elements
  • Definition 2.8: Dowker Complexes
  • Theorem 2.9: Dowker Duality dowker1952homology
  • ...and 87 more