The Flood Complex: Large-Scale Persistent Homology on Millions of Points

2025-09-26

AI Summary

Computing persistent homology (PH) for large-scale Euclidean point clouds remains intractable due to the exponential growth of the Vietoris–Rips complex, severely limiting scalability. Method: We propose the Flood complex—a theoretically sound simplicial complex combining advantages of Alpha and Witness complexes. Built upon Delaunay triangulation, it employs a radius-driven “flood fill” mechanism to dynamically select simplices, achieving linear space complexity while preserving topological correctness. The method supports sparse sampling and GPU-accelerated parallelization. Results: Experiments demonstrate efficient computation of 2D PH on million-point 3D point clouds. In geometrically and topologically complex object classification tasks, our approach significantly outperforms existing PH-based methods and point-cloud neural networks. This work constitutes the first empirical validation of feasible and effective large-scale topological feature learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Humans and AI: Crowd Sourcing and Human Computation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider the problem of computing persistent homology (PH) for large-scale Euclidean point cloud data, aimed at downstream machine learning tasks, where the exponential growth of the most widely-used Vietoris-Rips complex imposes serious computational limitations. Although more scalable alternatives such as the Alpha complex or sparse Rips approximations exist, they often still result in a prohibitively large number of simplices. This poses challenges in the complex construction and in the subsequent PH computation, prohibiting their use on large-scale point clouds. To mitigate these issues, we introduce the Flood complex, inspired by the advantages of the Alpha and Witness complex constructions. Informally, at a given filtration value $rgeq 0$, the Flood complex contains all simplices from a Delaunay triangulation of a small subset of the point cloud $X$ that are fully covered by balls of radius $r$ emanating from $X$, a process we call flooding. Our construction allows for efficient PH computation, possesses several desirable theoretical properties, and is amenable to GPU parallelization. Scaling experiments on 3D point cloud data show that we can compute PH of up to dimension 2 on several millions of points. Importantly, when evaluating object classification performance on real-world and synthetic data, we provide evidence that this scaling capability is needed, especially if objects are geometrically or topologically complex, yielding performance superior to other PH-based methods and neural networks for point cloud data.

Problem

• Computing persistent homology for large-scale point cloud data efficiently
• Overcoming computational limitations of Vietoris-Rips complex construction
• Enabling topological analysis on millions of points for machine learning

Innovation

• Flood complex uses Delaunay triangulation on subsets
• It efficiently computes persistent homology via flooding
• Enables GPU parallelization for millions of points