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$δ$-core subsampling, strong collapses and TDA

Elias Gabriel Minian

TL;DR

Persistent homology on large Vietoris-Rips filtrations is computationally demanding. The paper introduces the $δ$-core subsampling, a topology-preserving reduction based on strong collapses that removes dominated points to dramatically reduce the number of simplices while keeping global and local features. It provides formal development, uniqueness up to $δ$-equivalence, and complexity bounds, plus extensive experiments showing improved persistence diagram fidelity and substantial speedups over FPS and witness-based subsampling, with public code. This approach makes topology-aware PH scalable to large and heterogeneous datasets, enhancing the practical impact of topological data analysis.

Abstract

We introduce a subsampling method for topological data analysis based on strong collapses of simplicial complexes. Given a point cloud and a scale parameter $δ$, we construct a subsampling that preserves both global and local topological features while significantly reducing computational complexity of persistent homology calculations. We illustrate the effectiveness of our approach through experiments on synthetic and real datasets, showing improved persistence approximations compared to other subsampling techniques.

$δ$-core subsampling, strong collapses and TDA

TL;DR

Persistent homology on large Vietoris-Rips filtrations is computationally demanding. The paper introduces the -core subsampling, a topology-preserving reduction based on strong collapses that removes dominated points to dramatically reduce the number of simplices while keeping global and local features. It provides formal development, uniqueness up to -equivalence, and complexity bounds, plus extensive experiments showing improved persistence diagram fidelity and substantial speedups over FPS and witness-based subsampling, with public code. This approach makes topology-aware PH scalable to large and heterogeneous datasets, enhancing the practical impact of topological data analysis.

Abstract

We introduce a subsampling method for topological data analysis based on strong collapses of simplicial complexes. Given a point cloud and a scale parameter , we construct a subsampling that preserves both global and local topological features while significantly reducing computational complexity of persistent homology calculations. We illustrate the effectiveness of our approach through experiments on synthetic and real datasets, showing improved persistence approximations compared to other subsampling techniques.

Paper Structure

This paper contains 7 sections, 3 theorems, 3 equations, 8 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Strong collapses and cores of simplicial complexes
  3. Persistent homology
  4. The $\delta$-core subsampling
  5. Experimental results
  6. Conclusions
  7. Conclusions

Key Result

Proposition 2.4

If $K\hbox{$\ \searrow \! \! \! \! \searrow \! \! \! \! \ \ \ $} L$ then $K$ simplicially collapses to $L$ and, in particular, the inclusion $i: L \hookrightarrow K$ is a strong deformation retract. A simplicial retraction $r:K\to L$ which is a homotopy inverse to $i$ is defined by iterating the ret

Figures (8)

  • Figure 1: An elementary strong collapse.
  • Figure 2: $\delta$-core subsamples for different values of $\delta$.
  • Figure 3: Points in the $3$-dimensional cube: Comparison of persistence diagrams. Original sample and $\delta$-cores with $\delta=0.05$ and $0.3$.
  • Figure 4: Points in the cube: Comparison of persistence diagrams. Original sample, $\delta$-core with $\delta=0.25$, and FPS.
  • Figure 5: 2000 points sampled on the 2-torus in $\mathbb{R}^3$ in grey. The $\delta$-core for $\delta=1.4$ in blue (1527 points).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 2 more