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

TL;DR

The paper tackles the challenge of extracting the top-K intervals from the exterior-power persistence barcode B(Λ^i M) without full enumeration. It develops a birth-anchored, rank-grouped structural decomposition that splits B(Λ^i M) into monotone per-anchor streams with explicit multiplicities, enabling a best-first algorithm with running time O((M+K) log M) for fixed i. It proves the Top-K length vector is 2-Lipschitz under bottleneck perturbations and establishes an Ω(M log M) preprocessing lower bound, confirming optimality in the comparison model. Empirically, the method yields speedups over full enumeration in high-overlap scenarios, making higher-order persistence scalable for data analysis and ML pipelines, while providing robust, stable summaries for downstream use.

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.

Theorems & Definitions (16)

• theorem 1: Exterior-power interval calculus
• proof
• theorem 2: Stability of exterior powers
• proof
• lemma 1
• proof
• proposition 1
• proof
• theorem 3: Anchored rank--grouping
• proof