Approximating Persistent Homology for Large Datasets

TL;DR

This paper tackles the computational bottleneck of persistent homology on massive datasets by proposing a statistically principled multiple-subsampling framework. It extends the approach to three representations of persistence: Hölder continuous vectorizations (HCVs) of persistence diagrams, persistence measures, and persistence diagrams themselves, deriving nonasymptotic convergence rates via the Law of the Iterated Logarithm in Banach spaces and stability-based bias analyses. The authors provide practical guidance for parameter tuning, validate the theory with simulations on synthetic geometries, and demonstrate applications including permutation tests on Poincaré embeddings of large lexical databases. The work enables scalable, interpretable topological data analysis by delivering both theoretical guarantees and actionable procedures for real-world datasets. The methods hold promise for broader use in shape analysis, lexical topology, and clustering where exact persistent homology is infeasible.

Abstract

Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in diverse domains. Despite its widespread use, persistent homology is simply impossible to compute when a dataset is very large. We study a statistical approach to the problem of computing persistent homology for massive datasets using a multiple subsampling framework and extend it to three summaries of persistent homology: Hölder continuous vectorizations of persistence diagrams; the alternative representation as persistence measures; and standard persistence diagrams. Specifically, we derive finite sample convergence rates for empirical means for persistent homology and practical guidance on interpreting and tuning parameters. We validate our approach through extensive experiments on both synthetic and real-world data. We demonstrate the performance of multiple subsampling in a permutation test to analyze the topological structure of Poincaré embeddings of large lexical databases.

Figures (11)

• Figure 1: Convergence rate verification for mean persistence measure on the Torus (point cloud of $N = 50,000$ points).
• Figure 2: Convergence rate verification for mean persistence measure on the Sphere (point cloud of $N=20,000$ points).
• Figure 3: Training of Poincare embedding of Mammal subtree of WordNet at different epochs. We select 20 samples for visualization.
• Figure 4: Illustration of filtrations and persistence diagrams. (a) Sublevel set filtration of the distance function to a figure eight with Gaussian noise. (b) Vietoris--Rips filtration of samples from a figure eight with Gaussian noise.
• Figure 5: Illustration of Fréchet means of persistence diagrams and mean persistence measures for varying sample sizes. The left panel is the true persistence diagram of the sample set $\mathcal{X}$ computed from the Annulus. The top row shows the Fréchet mean of persistence diagrams of subsampled sets of different sizes. The bottom row shows the mean persistence measures of persistence diagrams of subsampled sets of different sizes.

Theorems & Definitions (53)

• Definition
• Proposition 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6
• Proposition 7
• Lemma 8
• Lemma 9