The Flood Complex: Large-Scale Persistent Homology on Millions of Points
Florian Graf, Paolo Pellizzoni, Martin Uray, Stefan Huber, Roland Kwitt
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 $r\geq 0$, 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 assessing object classification performance on both 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 to PointNet-like models.