Scaled two parameter homology reveals detailed data shapes fast
The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology
Computational GeometryMachine Learning
Summary
Finding patterns and shapes inside complex data helps make sense of it. However, methods that look at two features at a time often struggle with big data. The authors present a way to efficiently find these patterns by using a method called the sublevel Flood bifiltration. It’s designed to work faster on large datasets and is good at handling data where density matters, such as in time series. This approach keeps important theoretical qualities while making the process more practical for real applications.
What this means in practice
- •For data engineers: Handle large-scale datasets with two-feature shape analysis for improved pattern recognition workflows.
- •For signal processing teams: Enhance classification of time series by efficiently incorporating density-sensitive shape information from data.
Authors
Mattéo Clémot, Julie Digne, Julien Tierny
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.