Title: The Flood Complex: Large-Scale Persistent Homology on Millions of Points
Venue: NeurIPS 2025
Accepted: yes
Keywords: persistent homology

topological data analysis
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 $r\geq 0$, the Flood complex contains all simplices from a Delaunay triangulation of a small subset of a point cloud $X$ that are fully covered by the union of 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 GitHub: https://github.com/plus-rkwitt/flooder


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel filtration for point clouds the *Flood Complex*,\nthat approximates the well known Alpha complex, by using a smaller point cloud,\ne.g., a subsample, while still preserving geometric information from the\noriginal point cloud.\n\nThe paper provides theoretical approximation guarantees, as well as details to\ncompute an approximation and relaxation for a GPU implementation, which further\nmotivates this construction.\nIt finishes with an experimental section, showcasing the efficiency of the\nalgorithm as well as the statistical performances of the homology of this\nfiltration, which shows the practical the informativeness of this approach.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n - This filtration is intuitively reasonable.\n - The approximations guarantees are a must-have.\n - The relaxation to a GPU-friendly approach is very positive. This kind of\n optimization making TDA practical is very important for general applications\n in modern machine learning.\n - The proofs are sound.\n - This paper is very well written. Figure 1 gives a good intuition.\n - Experiments. The gain w.r.t. the full Alpha complex is very motivating.\n\n\n## Weaknesses\n - Choice of $L$. Although the paper treat it by refering to fpsample, there\n are no guarantees given w.r.t this choice. There are some existing work on\n Witness complex that recovers some simplices of the delaunay complex for\n instance.\n - Choice of $L$ 2. The accuracies seem to reflect that this choice is not too\n impactful, but the paper do not provide an analysis on this choice. It would\n be interesting to see what (theoretically, and in practice) happens when we\n choose $L$ as a random (possibly non-uniform) subsample of $X$, and under some\n reasonable assumptions on the sampling measure of $X$. Such rates of\n convergence would allow to understand the tradeoff that we make when choosing\n the size of $L$.\n - Code. I installed the code and had issues with fpsample, so I couldn't run\n it. I manually checked the code, and it looked simple enough so that I could\n quickly read it. This is not a terrible issue, as this approach seems very\n easily implementable.\n - Code2. Unless I missed something, the code for the experiments is not given.\n - This is not fully on the GPU since a Delaunay complex is still needed in the\n end. This can be problematic in higher dimensions or if $L$ is too large (but\n is still better than the full Alpha complex).\n - It seems to me that this filtration is differentiable w.r.t. the initial\n point cloud, but this is not the case for its approximation on GPU.\n - Experiments. I was expecting to see bootstrap methods (e.g., Expected\n persistence diagrams / landscapes) or witness / sparse filtrations to be in\n the main body of the paper are they are natural competitors.\n - Experiments 2. I feel that they are too few datasets in the experiments. \n For instance, the only time the alpha complex could run was on an experiment\n in which it looses against the (less geometric informative) Flood complex.\n - Scope. I feel that the main contribution of this paper is in the field of\n computational geometry. I don't know how pertinent this work is to the NeurIPS\n audience.\n - Theorem 3. It would be very nice if this was true for any Euclidean space.\n\nQUESTIONS:\n- Can you comment on why the Flood complex performs better than the (more\n geometrically accurate) Alpha complex (when Alpha is computable) ?\n - Can you comment on the stability of the approach w.r.t. $L$ ?\n - Up to some proper quotient / filtrations, I feel that there is  a link between the\n perisistence image (c.f., [Persistent Homology for Kernels, Images, and\n Cokernels, 2009]) of the inclusion map of the Delaunay complex on $L$ into the\n alpha complex of $X$ and the persistent homology of your filtration.\n I wasn't able to formalize this, but I'm interested if you have the answer of\n this. This would also allow to define the Flood complex PH on other\n filtrations, e.g., Rips.\n - In Lemma 6. Isn't there a better (deterministic / entrop maximizing) way of\n putting the points in a simplex to better optimize this bound ? A similar\n statement can be said for (a,b)-standard measures [cuevas, 2004], but that's\n not very important.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes the Flood complex to efficiently compute Persistence homology on large-scale point cloud data. Flood complex on a set of landmarks L contains all such simplifies of the Delaney complex on L that are covered by balls of radius r originating from data points X. Theoretical results shows that flood complex is an approximation of Alpha complex on X (alpha(X)), contains less simplifies than alpha(X), and due to efficient GPU implementation of computing filtration values it results in scalable and faster alternatives to alpha(X).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The Flood complex is efficient to compute and scales to large-scale point clouds. \n2. The Flood complex results in better point cloud classification accuracy than the  alpha complex on the landmarks in a reasonable time. \n\nWeaknesses:\n1. **Limited consideration of Baselines:** Baselines are limited to alpha complex and sparse rips complex only. There several recent subsampling/landmark based PH approximation methods such as [1,2] that construct filtration for scalability, yet they have not been considered in the experiments. \n\n[1] Adaptive approximation of persistent homology, Journal of Applied and Computational Topology, 2024\n\n[2] Topological Data Analysis with $\\epsilon$-net Induced Lazy Witness Complex, International Conference on Database and Expert Systems Applications, 2019\n\n2. **Sensitivity analysis:** The authors do not study the sensitivity of Flood Complex with  respect to the choice of Landmarks. \n\n3. **Tradeoff between \\#landmarks and Approximation quality:** It is important to understand the tradeoff between the number of landmarks and PH approximation quality. Although the authors empirically studied it in Figure 4(d), the paper does not give a recipe for how many landmarks are enough for some user-specified approximation quality in terms of bottleneck distance. As a result, it is hard to determine how many landmarks someone should use when they deploy Flood complex for their own datasets. \n\n4. **Experimental results:** \n\n(a) The authors should report the tradeoff between #Landmarks and Acc. of Flood complex across various datasets in Table 1. \n\n(b) In Table 1, the improvement in Acc. (Due to Flood Complex) is negligible compared to Alpha+. \n\n(c) Similarly, in Figure 3, it seems the accuracy of alpha(X) with 100k landmarks is almost the same as Flood complex with 500k landmarks. Does that mean that if someone uses alpha(X) with more landmarks, one could obtain better results than Flood complex? Perhaps the author can extend the x-axis to 500k points to clarify this. \n\nd) In Figure 4, explain why Alpha is slower than the Flood complex? Is it due to using Gudhi, which is a CPU-based implementation in contrast to the Flood complex, which is GPU-based? Or is it because the Alpha complex is computed on the entire point cloud rather than on landmarks? Also, why is the dimension 2 column in Fig. 4(b) missing? \n\n5. **Better clarification on the role of GPU in Flood complex construction:** Referring to the supplementary material, I noticed the flood complex construction is not entirely based on GPU, the authors still need to (1) select landmarks on CPU and (2) construct alpha complex on the landmark on CPU. Hence, the authors should clarify the reason for the faster running time of the Flood complex and exactly what rolethe  GPU plays there.\n\nQUESTIONS:\n1. In Figure 2, why Flood(X, L) total time the same as the Construction time? Does that mean the PH computation time is negligible? The same question for Fig. 4(a). Could you provide a more detailed breakdown of the total time in a table? \n\n2. Can the Flood complex be extended to non-Euclidean data, e.g., graphs?\n\n3. What is the computational complexity of Flood filtration construction?\n\nI am willing to adjust my score based on a satisfactory resolution of Weaknesses 1-4 with experiments.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on the central problem of the computational expense of persistent homology.  The paper proposes a new complex that is able to handle large scale point clouds and thus make the computation of persistent homology on some datasets that have so far not been possible.  Some theoretical guarantees are given and a nice, complete experimental study is carried out.\n\nWhile I would not classify the contribution of the work to be under \"AI\", it does make persistent homology a viable tool to then be used to study problems in AI.\n\nSTRENGTHS AND WEAKNESSES:\nThe problem that the paper studies is undoubtedly one of the most important problems in computational persistent homology and, in my view, one of the main limitations of the approach in applications: persistent homology simply does not scale well and there are serious problems with applying it to datasets that are common in current application areas and other fields, and definitely machine learning.  The contribution of this submission has the potential to overcome this and expand the potential and applicability of persistent homology as a tool.\n\nA limitation of the work that I see is that it still relies on considering smaller subsets of the data, which is a natural/classical approach and takes away from the originality but nevertheless, the contribution of the work is definitely very important.  Additionally, proposing another complex is not very original, but again, nevertheless, a solid way to workaround the problem.  However, the biggest weakness I see and most problematic aspect is that it appears that the appendix was not submitted and so I could not check details (and would have liked to), which I see as quite a serious oversight.  This is the main reason for my borderline assessment of the paper.\n\nQUESTIONS:\nSome comparisons are carried out wrt the alpha and witness complexes, for example, which I suppose provides a satisfactory (but somewhat underwhelming) \"guarantee\" of the performance of the method, but an important outstanding question that does not seem to be addressed is the correctness of the result, more specifically, the correctness of the algorithm (which was not detailed in the submission, I imagine that it probably appeared the appendix which was not submitted).  In the experiments, persistent homology was computed on massive datasets where it was to date not possible: how can we be sure that the resulting barcodes are indeed the correct barcodes?  If correctness cannot be proven, then some theoretical results on convergence of the method to the \"true\" barcode is a really important missing theoretical result.  Are there details on this in the missing appendix?  If so, they should be moved to the main body of the paper.\n\nLIMITATIONS:\nLimitations were discussed briefly in the main body of the paper, but given the potential impact of the work, I would like to see a more honest, sober analysis and ablation study of the contribution.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 1, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper introduces a novel construction called the Flood Complex, designed for efficient computation of PH on large-scale Euclidean point clouds. The key idea is to construct a filtered simplicial complex over a small set of landmark points by checking whether the convex hulls of Delaunay simplices are fully “flooded” (i.e., covered) by balls centered at the full dataset. This construction approximates the Alpha complex, while being orders of magnitude faster and scalable to millions of points. The authors provide theoretical guarantees (stability, approximation bounds), GPU-accelerated implementation, and extensive experiments showing state-of-the-art performance on both synthetic and real-world classification tasks.

STRENGTHS AND WEAKNESSES:
Strengths:
+ The Flood complex is a creative and non-trivial extension of Alpha/Witness complexes, enabling scalable topological analysis on massive point clouds.
+ It tackles one of the central bottlenecks in Topological Data Analysis: applying PH at scale.

Weaknesses:
- In fact, I am completely unfamiliar with this direction. So, I don't know if it has any disadvantages.

QUESTIONS:
Please see my comments in the part of Strengths and Weaknesses.

LIMITATIONS:
Maybe yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02