Higher-order Persistence Diagrams

TL;DR

This paper introduces higher-order persistence diagrams that encode containment relations among persistence intervals, enabling more faithful structural comparison and interpretability in topological data analysis.

Contribution

It presents a recursive construction of higher-order diagrams, a harmonic analysis approach for efficient evaluation, and demonstrates speedups in experiments.

Findings

Substantial speedups over explicit aggregation methods.

Preserves interval-level structure for better interpretability.

Enables comparison and aggregation directly on persistence diagrams.

Abstract

Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201.