Flood Complex in Persistent Homology

• Flood complex is a filtered simplicial complex that uses Delaunay triangulation on a landmark subset and flooding from the full dataset to capture fine topological features.
• It provides theoretical guarantees such as bottleneck stability and homotopy equivalence while approximating classical Alpha complexes with controlled error margins.
• By reducing simplex counts and employing GPU-parallel computation, the Flood complex scales to millions of points, outperforming Vietoris–Rips and standard Alpha complexes.

The Flood complex is a filtered simplicial complex construction for large-scale persistent homology (PH) computation on Euclidean point clouds. It is designed to address the computational and representational bottlenecks of classical topological data analysis (TDA) structures such as the Vietoris–Rips and Alpha complexes, enabling PH computation and subsequent machine learning workflows on point clouds with millions of points. The Flood complex adapts Delaunay triangulation principles and “floods” the triangulation using balls defined by the full dataset, yielding favorable theoretical guarantees, substantial reduction in simplex counts, GPU-parallel computation, and improved discrimination of geometric/topological features in downstream tasks (Graf et al., 26 Sep 2025).

1. Construction of the Flood Complex

Given a finite point cloud $X \subset \mathbb{R}^d$ and a landmark subset $L \subset X$ (typically much smaller than $|X|$), the process begins with the Delaunay triangulation $\operatorname{Del}(L)$. At filtration scale $r \geq 0$, the Flood complex is defined as: $$\operatorname{Flood}_r(X,L) = \left\{ \sigma \in \operatorname{Del}(L) : |\sigma| \subseteq \bigcup_{x \in X} B_r(x) \right\},$$ where $|\sigma|$ is the convex hull of the simplex $\sigma$ and $B_r(x)$ is the closed ball of radius $r$ centered at $x$. Each simplex $\sigma$ enters the filtered complex at the minimal $r$ such that all its points are “flooded” (i.e., $$f(\sigma) = \max_{p \in |\sigma|} \min_{x \in X} d(p, x)$$).

Unlike the Alpha complex—whose simplices and filtration are defined strictly on $L$, potentially ignoring fine structure in $X$—the Flood complex leverages Delaunay simplices from $L$ but assigns filtration values anchored to $X$. This hybrid construction enables the complex to efficiently encode fine topological features determined by the full dataset, using a small triangulation.

2. Theoretical Properties

Several key theoretical properties characterize the Flood complex:

• Bottleneck Stability: For two point clouds $X$ and $X'$, with the same landmark set $L$, the persistence diagrams of their corresponding Flood complexes satisfy

$$d_B\left(\operatorname{dgm}_i(\operatorname{Flood}(X,L)), \operatorname{dgm}_i(\operatorname{Flood}(X',L)) \right) \leq d_H(X,X'),$$

where $d_B$ is the bottleneck distance between diagrams, and $d_H$ is the Hausdorff distance between point sets. This guarantees that the computed persistent features are insensitive to small perturbations in $X$.

• Homotopy Equivalence: When $L = X$ (i.e., all points are used as landmarks), the geometric realization of $\operatorname{Flood}_r(X,X)$ equals the union of balls $\bigcup_{x \in X} B_r(x)$, and thus is homotopy equivalent to the standard Alpha complex at that scale, as per the nerve theorem.
• Persistence Approximation: If the Hausdorff distance $d_H(X,L)$ is small, then the persistence diagrams satisfy

$$d_B\left(\operatorname{dgm}_i(\operatorname{Alpha}(X)), \operatorname{dgm}_i(\operatorname{Flood}(X,L))\right) \leq 2\, d_H(X,L),$$

so the Flood complex provides a controlled approximation to Alpha PH even when using a subset of landmarks.

3. Advantages over Vietoris–Rips and Alpha Complexes

• Combinatorial Efficiency: The combinatorial explosion intrinsic to Vietoris–Rips (where all $k$-simplices up to a diameter threshold are constructed) makes it prohibitive for large $|X|$. The Flood complex constructs at most $O(|L|^{\lfloor d/2\rfloor})$ simplices, as it is based on a Delaunay triangulation of $L$.
• Incorporation of Full Data Information: Compared to an Alpha complex built solely on $L$, filtration in the Flood complex remains sensitive to the geometry and topology of the full cloud $X$ via the flooding process, thus preserving fine persistence features lost in subsampling.
• GPU Parallelism: A GPU implementation is made feasible by “masking,” i.e., for each simplex, preselecting those $x \in X$ possibly close enough to affect flooding, enabling rapid parallel computation of simplex filtration values (via custom Triton kernels).

4. Scalability for Large-Scale Point Clouds

The Flood complex framework delivers PH computation in dimensions 0-2 on point clouds with 10–12 million points in practical timescales (seconds to minutes). Filtration assignments based on “flooding” keep memory usage low relative to the full Alpha complex and maintain computation within practical hardware resources even at extreme point cloud scales. This is essential for industrial PH applications and large-scale 3D shape analysis.

Construction	Full point cloud dependency	Number of simplices	Scalability (GPU)
Vietoris–Rips	Yes	Exponential in $|X|$	No
Alpha Complex	Yes (on $X$)	Large but manageable	Partial
Alpha (on subsample)	No (on $L$)	Smaller, less informative	Yes
Flood complex	Yes (on $X$ for flooding)	Like Alpha on $L$	Yes (with masking)

5. Experimental Validation and Applications

Experiments show that the Flood complex outperforms other scalable PH approximations (e.g., Alpha PH on subsampled data) and even neural approaches (such as PointNet++) in classification tasks where subtle geometric or topological intricacies matter, such as:

• Synthetic Data: On datasets like “swisscheese” (distinguishing number of holes) and “rocks” (porous structure analysis), Flood PH produces persistence diagrams with more accurate birth and death values and higher balanced classification accuracy.
• Real-World Data: On ModelNet10 and mechanical CAD components, Flood PH achieves strong results, with the difference to neural networks being most significant on highly intricate object classes (e.g., coral surface structure and porous rocks).

The key finding is that the “flooding” process enables the preservation of global and local topological descriptors necessary to distinguish classes that are difficult for simpler PH subsampling or for networks lacking explicit topological regularization.

6. Mathematical Formulation and Guarantees

• Complex Definition:

$$\operatorname{Flood}_r(X, L) = \{ \sigma \in \operatorname{Del}(L) : |\sigma| \subseteq \bigcup_{x \in X} B_r(x) \}$$

• Filtration Assignment:

$$f(\sigma) = \max_{p \in |\sigma|} \min_{x \in X} d(p, x)$$

• Stability (bottleneck):

$$d_B\left(\operatorname{dgm}_i(\operatorname{Flood}(X,L)), \operatorname{dgm}_i(\operatorname{Flood}(X',L)) \right) \leq d_H(X,X')$$

• Approximation to Alpha:

$$d_B\left(\operatorname{dgm}_i(\operatorname{Alpha}(X)), \operatorname{dgm}_i(\operatorname{Flood}(X,L))\right) \leq 2\, d_H(X,L)$$

These guarantees formalize both numerical stability and the loss bounds relative to idealized full-complex computations.

7. Implications and Significance

The Flood complex provides a scalable, stable, and information-rich method for extracting topological summaries from large-scale point clouds. In high-volume or high-resolution data regimes common in modern computer vision, remote sensing, and 3D scanning, it enables single-pass persistent homology computations that can be directly incorporated in machine learning pipelines (e.g., for object recognition or structural analysis) where the input geometry is both voluminous and topologically intricate. Its empirical superiority over both traditional PH and neural network baselines in several domains underscores the importance of scaling and geometric fidelity in TDA.

The Flood complex, by combining Delaunay-based triangulation, efficient flooding-induced filtration, and GPU parallelization, is a foundational advancement in the computational topology of large datasets, and is particularly well-adapted for persistent homology pipelines in modern geometric data analysis (Graf et al., 26 Sep 2025).