Higher-order Persistence Diagrams

2026-05-10

AI Summary

This work addresses a critical limitation in existing vectorization and kernel methods for topological data analysis, which neglect the rank-induced inclusion relations among persistence intervals, thereby distorting structural information and reducing interpretability. To overcome this, the authors propose high-order persistence diagrams that explicitly capture structural dependencies by recursively modeling interval inclusion relationships. They further introduce, for the first time, harmonic analysis and the zeta transform to implicitly aggregate high-order diagrams in the spectral domain, reducing computational complexity from quadratic to nearly linear. Experimental results demonstrate that the proposed method significantly outperforms explicit aggregation strategies on random network models, achieving superior efficiency and scalability while preserving structural fidelity.

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.

Problem

• persistence diagrams
• topological data analysis
• interval containment
• structural comparison
• interpretability

Innovation

• higher-order persistence diagrams
• topological data analysis
• zeta transform
• harmonic analysis
• persistence interval aggregation