The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology

2026-10-04

AI Summary

This study addresses the scalability limitations of multiparameter persistent homology on large-scale data by proposing the sublevel Flood bifiltration as an efficient approximation of the sublevel offset bifiltration. By extending the single-parameter Flood filtration to the biparameter setting, the proposed method integrates topological data analysis with bifiltration theory, effectively balancing topological stability and computational efficiency. The effectiveness and robustness of this approach are validated through classification tasks on both synthetic and real-world time series datasets. Ultimately, this work provides a scalable new pathway for extracting multiparameter topological features from large-scale data.

Abstract

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

Problem

• multiparameter persistent homology
• scalability
• topological data analysis
• large point sets

Innovation

• multiparameter persistent homology
• sublevel Flood bifiltration
• scalability
• topological data analysis
• stability