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

arXiv: https://arxiv.org/abs/2509.22432 · HTML: https://mlsanity.com/paper/2509.22432/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 54 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Rethinking Network Design and Local Geometry in Point Cloud: A Simple Residual MLP Framework](https://mlsanity.com/paper/2202.07123/) (2022)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [Topological Attention for Time Series Forecasting](https://mlsanity.com/paper/2107.09031/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complex](https://mlsanity.com/paper/2303.01501/) (2023)
- [The Morse theory of vCech and Delaunay complexes](https://mlsanity.com/paper/1312.1231/) (2013)

## Built on by (most cited first, 6 of 6)

- [Learning Significant Persistent Homology Features for 3D Shape Understanding](https://mlsanity.com/paper/2602.14228/) (2026)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [Prototype Selection Using Topological Data Analysis](https://mlsanity.com/paper/2511.04873/) (2025)
- [Learning Topological Representations of Protein Structure and Dynamics](https://mlsanity.com/paper/2606.14737/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

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 r≥ 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. Source code and datasets are available on https://github.com/plus-rkwitt/flooder.