[Paper Review] Approximating Persistent Homology for Large Datasets

Yueqi Cao, Anthea Monod|arXiv (Cornell University)|Apr 19, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper proposes using bootstrapped subsampling to approximate persistent homology for large datasets where direct computation is infeasible. By computing the mean of persistence diagrams from multiple subsamples, the method provides a statistically valid and convergent approximation to the true persistence diagram, with explicit convergence rates derived via Fréchet means and Wasserstein stability in random set theory.

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.

Motivation & Objective

• To address the computational infeasibility of persistent homology on very large datasets.
• To develop a statistically valid approximation of the true persistence diagram using subsampling.
• To establish finite-sample convergence rates for the mean of subsampled persistence diagrams to the true persistence measure.
• To preserve interpretability by returning results as persistence diagrams rather than vectorized forms.
• To extend the applicability of topological data analysis to large-scale real-world data, including point clouds and metric spaces.

Proposed method

• Adapts classical bootstrapping by drawing multiple smaller subsamples from a large dataset.
• Computes persistence diagrams for each subsample and takes their mean as a mean persistence measure.
• Uses Fréchet means in the space of persistence diagrams under the Wasserstein distance to define the mean.
• Applies convexity and stability properties of persistence diagrams, combined with random set theory, to derive convergence bounds.
• Employs bias–variance decomposition to control approximation error and guide subsample size and number.
• Validates results using synthetic and real datasets, including shape clustering on large point clouds.

Experimental results

Research questions

• RQ1 Can the mean of persistence diagrams from subsamples provide a consistent approximation to the true persistence diagram of a large dataset?
• RQ2 What is the finite-sample convergence rate of the mean persistence diagram to the true persistence measure?
• RQ3 How does the choice of subsample size and number of subsamples affect the approximation error?
• RQ4 Can this method preserve interpretability while enabling scalable topological analysis?
• RQ5 Does the approach generalize to non-Euclidean data types such as finite metric spaces and weighted graphs?

Key findings

• The mean persistence diagram computed from subsamples converges to the true persistence diagram at a quantifiable rate, with explicit bounds derived using random set theory.
• The method achieves convergence under optimal partial transport distance, with theoretical guarantees on bias and variance components.
• The approach maintains interpretability by returning results as persistence diagrams, avoiding lossy vectorization.
• Empirical validation on synthetic and real data, including 'Knot', 'Lock', 'Bearing', and 'Motor' point clouds, confirms strong performance in shape clustering.
• The method performs well on general finite metric spaces and weighted graphs, suggesting potential for broader generalization beyond Euclidean point clouds.
• The framework enables practical topological analysis on datasets too large for direct persistent homology computation.