Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability

TL;DR

This paper introduces a new algorithm for efficiently extracting the top K longest intervals in exterior power persistent homology, with proven stability and practical speedups for large datasets.

Contribution

It provides a structural decomposition theorem and a best-first algorithm for exterior power persistent homology, enabling scalable analysis of large datasets.

Findings

The algorithm achieves significant speedups over full enumeration.

The top-K length vector is 2-Lipschitz under barcode perturbations.

Experiments confirm the theoretical speedups in high-overlap cases.

Abstract

Exterior powers play important roles in persistent homology in computational geometry. In the present paper we study the problem of extracting the $K$ longest intervals of the exterior-power layers of a tame persistence module. We prove a structural decomposition theorem that organizes the exterior-power layers into monotone per-anchor streams with explicit multiplicities, enabling a best-first algorithm. We also show that the Top-$K$ length vector is $2$-Lipschitz under bottleneck perturbations of the input barcode, and prove a comparison-model lower bound. Our experiments confirm the theory, showing speedups over full enumeration in high overlap cases. By enabling efficient extraction of the most prominent features, our approach makes higher-order persistence feasible for large datasets and thus broadly applicable to machine learning, data science, and scientific computing.