Node-Level Topological Representation Learning on Point Clouds

Thank your very much for your review and your suggestions! Based on the reviews, we have uploaded a revised version of our paper. We will reply to your comments below.

1. Regarding the hyper-parameters, such as $\gamma$ on line 310 and $\delta$ on line 342, there are some concerns about robustness and the way to determine them. Although the authors test the robustness of the hyper-parameters in Appendix D, these results are limited to the topological clustering dataset. It remains uncertain if they are effective on more practical data.

The real-world datasets of the paper showed that TOPF with the default parameters performed well. In addition, we have added an experiment on points sampled from the high-dimensional conformation space of cyclooctane, see Figure 11, where TOPF again performed well. Thus we believe that we have reasonable hope for the design and the hyperparameter selection of TOPF to perform well on a wide variety of unseen datasets. (This is partly because hyperparameters like $\delta$ and $\gamma$ only fine-tune the implemented heuristic/formula for choosing thresholds.) Furthermore, we also give intuition and guidance on how to select the optimal hyperparameters in Appendix D.1. While we acknowledge the concern of the reviewer, we believe we have done all possible reasonable steps to ensure that the hyperparameter selection is sufficiently robust.

2. The validity of the heuristics used in the paper should be further investigated. For example, the way to choose the points from persistence diagram in line 294, the strategy to give the topological information to each node in 342, and the process to aggregate the multiple information in line 350. These approaches should be validated with realistic datasets.

We thank the reviewer for this comment. In addition to the 11 datasets of the original paper, we have introduced the high-dimensional cyclooctane experiment (Figure 11), analysed the performance of TOPF with decreasing sampling density in Figure 15, and investigated the performance of TOPF on high-dimensional latent spaces of VAEs (Figure 13). Although more datasets are always good and we understand the concern of the reviewer, we have already validated TOPF on a wide range of possible topologically structured datasets.

3. Since various techniques are employed in the paper, it would be desirable to conduct an ablation study. Specifically, it should be tested on simple data how the effectiveness of features changes with or without the Hodge Laplacian.

Thank you for this suggestion, we have run an ablation study on using the Hodge Laplacian to project the homology generators into the harmonic space in contrast to directly working with the homology generators:

This shows that in general, skipping the step with the Hodge Laplacian decreases the performance of TOPF by a wide margin. The exception to this are SphereInCircle or 2Spheres2Circles, where the two methods have similar performance. In these two last examples, points are directly sampled from a manifold, and thus the loops and spheres are very "thin". Thus, the homology generator is already a good approximation of all the simplices responsible for a topological feature. In general however, this is not the case, necessating the use of Differential Geometry and the Hodge Laplacian. We will add a discussion of this to the paper.

4. The intent of the experiments using VAE is unclear. [..] Additionally, it is unclear there are any implication for any specific applications.

The goal of the VAE experiment was to show that it is possible for TOPF to detect topological structure in the latent space of variational autoencoders. This of course does not imply that the latent space of every VAE has a topological structure. However, we still believe that the experiment serves as a proof-of-concept for an interesting possible application of TOPF. As discussed above, we have now added a discussion of a high-dimensional latent space of a VAE as well. We will try to make this more clear in the paper.

It is difficult to understand what is meant by “topological structure inherent in the sample space,” as described in Figure 11.

Thank you for this remark! The image patches sampled from the image were sampled across the sides of two squares and a line connecting these squares. This structure induces a similar topology in the sample spaces, which then gets embedded into the latent space and detected by TOPF. We have added a more detailed description on this in the caption of Figure 11.

1. Why did you use VR filtration for n>3? In higher dimension, doesn’t the computation cost of the persistent homology increase since the number of simplices in the VR complex increase?

As is customary in TDA, we compute the persistent homology in dimensions 0, 1, and optionally 2 in our experiments. While computation of higher-dimensional homology is straightforward from a theoretical perspective, there are two separate reasons not to do this in practice:

1. The manifold hypothesis [1] suggests that even in high-dimensional ambient spaces, the data lies on a low-dimensional submanifold, in practice often of dimension 1 or 2. A well known fact from topology is that k-dimensional manifolds will have homology in degrees only up to their intrinsic dimension independent of the ambient dimension. A wealth of literature in TDA seems to support this assumption.
2. Computing persistent homology in degrees above 2 is computationally very prohibitive.

Thus, we keep the maximum homology degree fixed even for high-dimensional spaces. The computational complexity of computing the homology of a VR filtration is only depending on the number of points and the homology dimension and is independent of the ambient dimension. (Because the VR filtration operates on the distance matrix.) On the other hand, constructing the alpha-filtration depends on performing a Delaunay triangulation, which is computationally infeasible for high-dimensional spaces. Hence, the VR filtration is computationally optimal for high-dimensional spaces, even though it has more simplices. (But in order to compute k-dimensional homology, it suffices to consider the simplices of degree up to k+1, which can be far smaller than the ambient dimension.)

We will try to explain this reasoning more clear in the paper.

[1] Fefferman, Charles, Sanjoy Mitter, and Hariharan Narayanan. "Testing the manifold hypothesis." Journal of the American Mathematical Society 2016.

2. Is it possible to effectively use TOPF for machine learning tasks other than clustering, such as point cloud classification? Additionally, is quantitative evaluation feasible for such tasks?

Informed by the effectiveness of topological point cloud level features in Machine Learning applications we believe TOPF has the potential to be useful for downstream machine learning tasks. However, because not all point clouds possess topological structure measurable by homology and topology is often not the only thing we care for, we suggest that TOPF will be most effective when used in conjunction with other more geometric point features or other features form the point level. However, because topological features are different from other existing features and thus integrating TOPF into existing ML pipelines would require a new or strongly modified architecture, we believe that this is an exciting topic for future work.

3. How did you choose the dimension of the persistent homology in the experiments?

For the experiments in the TCBS, we set the persistent homology dimension to the maximum possible dimension permitted by the ambient dimension, which was 1 and 2 respectively. (It is a fact from algebraic topology that a closed submanifold of an $n$-dimensional manifold can have homology only in degree up to $n-1$)

For the experiments with real data, we use prior knowledge on the topological structure we were looking for to set the homology dimensions: For the NALCN channelosome and GroEL proteins, we wanted to detect the loops/1-dimensional holes in the protein structure, and looked at dimension 1. For the Cys123 (with the added convex hull), we looked for protein pockets and thus only looked at dimension 2.

For the VAE latent space embedding experiment, we looked for loops informed by the sampling strategy, and thus set the dimension to 1. For the Lorentz attractor data set, we again looked for loops around the non-attractive fixed points and thus used a homology dimension of 1. For the newly introduced cyclooctane dataset, we did not have prior information and thus set $d=2$.

We assume that in many use cases, we will already know for what kind of topological features we will be looking for. However, we can always run TOPF with a larger maximum homology dimension and TOPF auto-selects the relevant dimensions by comparing the life times of features across dimensions. (Which is of course more prone for error than using prior knowledge).

Your following statement is not correct:

Hence, the VR filtration is computationally optimal for high-dimensional spaces, even though it has more simplices.

VR filtration may contain $O(n^{k+1})$ simplices while computing $H_k$, where $k$ = maximum homology degree. For a large number of points $n$ (e.g., 100K points), it becomes computationally infeasible to compute VR homology features in commodity machines (with, say < 32 RAM), even for k = 2.

Thank you for your comment! We will reply to your other comments in more detail later, but will first address your concern above. First of all, thank you very much for reading our responses to the other reviewers! We appreciate this!

We wrote:

Hence, the VR filtration is computationally optimal for high-dimensional spaces, even though it has more simplices.

We could have worded this more carefully. In the paragraph we were talking about $\alpha$-filtrations vs VR filtrations. What we meant with the above cited sentence is that the VR-filtration is computationally optimal in comparison to the $\alpha$-filtration for high-dimensional spaces. The $\alpha$-filtration needs to compute a d-dimensional Delaunay triangulation, which has an exponential runtime in the dimension of the ambient space. The VR filtration on the other hand does not depend on the ambient dimension. Thus, in the setting of high-dimensional spaces, the VR filtration is computationally optimal compared to the $\alpha$-filtration. As you noted, this does not mean that the VR filtration is feasible for a very high number of points, it is not. In these cases we either have to

1. Perform farthest-point/random downsampling as employed in TOPF,
2. Pick a maximal $\epsilon$-radius for the VR-filtration. This can greatly decrease the number of simplices in the VR filtration, but it is not immediately clear how to choose $\epsilon_\text{max}$. However, this would work well with the rest of the TOPF pipeline,
3. Pick a different filtration type which might be more suited for the data (Like sparse VR). The only downside of this is that for other filtration types there are fewer existing implementations, im particular very few libraries compute the homology generators (instead of the cohomology generators), which work better for TOPF. This is why we stick to $\alpha$-filtrations and VR filtrations in our proposition of TOPF. We note that the TOPF pipeline is independent of the underlying filtration, and is thus very flexible for generalisation to other types of filtrations.

We hope to have addressed your concern and provided enough context as to how to interpret the above claim. We apologise if we caused confusion. Thank you again for your remark!

Thank you very much for taking the time to read our paper and for your review! We have uploaded an updated version of the paper based on the feedback we received, and will reply to your points below:

1. Fig. 1 doesn't help understand the overall and core design of the TOPF. The authors are highly suggested re-drawing this part. From my personal perspective of view, this figure should consist the motivation and the overview of the proposed methods.

Thank you for this suggestion! We have now added a brief explanation for the bars in step 1, added arrows to help readability, and improved the separation between the applications/motivation and the remainder of the method. We want to emphasise that the Figure 1 provides an overview over the method, providing illustration for input and output and the four main steps as described in section 3. Furthermore, we highlight potential applications which serve as the motivation for TOPF. We hope to have addressed your concerns. However, if you still have concrete suggestions for improving the figure, please let us know!

2. For the experimental part, point cloud matching and registration are typical tasks that require good representation point clouds. I would suggest the authors add comparisons to those related methods, like the unsupervised ones (PPF-FoldNet, ECCV 2018 & WSDesc, TVCG 2022).

Thank you for the excellent suggestion! Benchmarking against unsupervised (or rather weakly supervised methods) seems indeed like a substantial strengthening of our claims. We have added benchmarks agains WSDesc (Pretrained on 3DMatch), as suggested by you!

While WSDesc has a better performance than PointNet on the 3D datasets, TOPF still outperforms it by a wide margin on the more challenging sets of the TCBS. In addition to the 32-dimensional features produced by WSDesc, we have added a feature channel for the normalised 3d coordinates, because this greatly improved the performance of WSDesc on TCBS.

The obtained results are very interesting: On the easiest 3D dataset, 2Spheres2Circles (with a large (>4000) number of points and uniform sampling), WSDesc matches the performance of TOPF with an ARI of 0.95. On Spaceship, WSDesc achieves an ARI 0.76 (TOPF: 0.92). Looking at the results directly, we interpret this as WSDesc struggling with the non-uniform sampling across the different features. Finally, on SphereinCircle, WSDesc achieves an ARI of 0.46. This is better than most other non-toplogical methods, but significantly worse than TOPF with 0.97. In our interpretation, this has to do with the sparseness of just 267 points.

On the 2D datasets (which we embedded into 3D) WSDesc performs very poorly, which might be expected because it was trained for 3D datasets. This shows that on the TCBS, TOPF is competitive even against weakly supervised methods which excel on current benchmarks.

In case you have any further questions, we will be happy to answer them! We are looking forward to the discussion and hearing whether you assessment of our paper has changed based on the rebuttal! Thank you again!

Dear Authors,

I appreciate your efforts and response a lot, thanks again! After throughly reading your reply and discussion with other reviewers, my rating towards this paper is still slightly negative. Just as reviewer hKZz said, as for a journal submission, this paper should get one major revision, but for conference, I cannot recommend accpet for the current version. Mainly due to the clarity in paper writing and figure drawing, and also because of the experimental settings. One major flaw could be the lack of strong baselines, like unsupervised point cloud representation methods (PPFFoldNet or WSDesc) and general feature extractors (DGCNN or KPConv). As a result, I will keep my initial rating.

Best,

YUvn

Thank you very much for your reply and reading our comments and the discussion! We will reply below.

as for a journal submission, this paper should get one major revision

We have uploaded a revised version of the paper, which addresses feedback by the reviewers. In particular, we have included experiments on sampling density and heterogeneity, experiments on high-dimensional data, included WSDesc as a strong benchmark, updated Figure 1, moved Theorem 1 to the appendix to have more time to speak about the experiments, and more. If interpreting this from a journal-submission perspective, this could be seen as the major revision. However, you are of course right to request as many major revisions as you deem fit.

Mainly due to the clarity in paper writing and figure drawing

We have already updated Figure 1 and tried to improve the clarity of writing. Without any detailed feedback, we sincerely do not know what you would want us to do. We believe Figure 1 in its current form is a good illustration of the method, as for example Reviewer 8pLm praised our illustrations. The same reviewer praised the presentation and the step-by-step description of section 3, while even reviewer smnw wrote that our paper is written clearly enough.

Despite this, you write that the clarity in paper writing and figure drawing is the main reason to reject our paper. We would thus ask you to either provide us with concrete feedback that we can act on or to reevaluate your assessment. We do not write this to invalidate your opinion, we would only like to understand whether there is anything concrete that we could change in our paper.

Thank you very much!

One major flaw could be the lack of strong baselines, like unsupervised point cloud representation methods (PPFFoldNet or WSDesc) and general feature extractors (DGCNN or KPConv)

Thanks to your feedback, we have included a benchmark against WSDesc in our revision (see the uploaded pdf), which we outperform. Does this address your concern?

We would furthermore like to add that one of the strengths of TOPF lies in the fact that it is not just a different neural architecture, but rather is founded on insights from algebraic topology, does not require a GPU or pretraining, and has interpretable output. Thank you again for your review!

We thank the reviewer for their review. We will reply to their comments below.

Lines 177-178: "the filtration value of a k-simplex in the α-filtration is the radius of its circum-k-sphere, which differs from its filtration value in the VR filtration". The concept of a "filtration value" appears here for the first time and is never defined in the paper, which leads to the major misunderstanding of persistence.

Thank you for raising this issue. We have added a sentence explaining that the value a simplex $\sigma$ first appears in the filtration is called its filtration value $\sigma$. I.e., for a filtration $F$, we have that $F(\sigma)=\inf (\varepsilon \in \mathbb{R}: \sigma\in F_\varepsilon)$.

We apologise, but what is the major misunderstanding of persistence the reviewer is referring to?

Even a non-expert can notice that n-th dimensional (?) homology class cannot be generated by 2-dimensional (?) simplices for n>2.

Thank you for pointing out this typo, it should read $n$-simplices instead of $2$-simplices as used throughout the proof. We have fixed this.

Also, Theorem 4.1(i) fails for n=1 and any 3 points on a unit circle that form a non-acute triangle.

We realise now that this confusion stems from our definition of alpha-complexes (Section 2: Vietoris--Rips and $\alpha$-complexes and Definition B.2 in our original submission) being slightly different from the definition of alpha complexes in the literature and current implementations (Including of the libraries TOPF uses). In order to avoid confusion, we have now renamed our original construction to "$\alpha^*$-complexes" (Definition B.2, and Section 2: Vietoris--Rips and $\alpha$-complexes), gave a definition of the literature version of $\alpha$-complexes (Definition B.4)and discussed the differences. We now formulate Theorem 4.1 in terms of the $\alpha^*$-filtration and give conditions for when the theorem holds for $\alpha$-complexes as well. We want to reiterate that our theorem is true in its current form, and was true in the original submission when using the definitions given in the paper.

There are infinitely many point clouds whose 1-dimensional persistence is empty. Look for a generic family of such clouds in J Applied and Comp Topology 2024.

We appreciate the reference, but we fail to see the relation to our paper. There are infinitely many point clouds whose $1$-dimensional persistence is non-empty, and a wealth of literature in TDA proves that these point clouds with topological structure appear in many applications, see our introduction for more references, or surveys like [1] or [2]. Clustering methods work only on data with a cluster structure, image classification methods only on images which are different to noise, and so on. Still all of these methods are valuable for the data they are designed for, and the same holds for TOPF.

[1] Elizabeth Munch. A user’s guide to topological data analysis. Journal of Learning Analytics, 2017

[2] Larry Wasserman. Topological data analysis. Annual Review of Statistics and Its Application, 2018.

Even Figure 3 in the paper demonstrates that only 3 points in the 1D persistence represent reasonable cycles, hundreds of others represent only noise.

We apologise, but we do not understand the concern of the reviewer here. In the cited example, our method is able to extract the three relevant persistent homology classes and build topological features using them while discarding the noise ones by considering the birth and death times of the persistence barcodes. This is exactly as persistent homology and our method was designed to perform. Figure 5b) shows that the output of TOPF represents meaningful topological structure.

Lines 514-520: "selection of the relevant features is a very hard problem" and "efficient computation of simplicial weights leading to the provably most faithful topological point features is an exciting open problem" The words "relevant" and "most faithful" should be rigorously defined, else anyone can claim that their features are "relevant" and "most faithful".

The reviewer seems to mistake our intention in the quotes above. The full quotes are

Limitations [...] Finally, selection of the relevant features is a very hard problem. While our proposed heuristics work well across a variety of domains and application scenarios, only domain- and problem-specific knowledge makes correct feature selection feasible.

Future Work [..] Furthermore, efficient computation of simplicial weights leading to the provably most faithful topological point features is an exciting open problem.

One of the reasons that selecting the relevant features is hard is that there is no unique definition of relevant features across different application domains. Thus, we conclude in the paper that while our heuristics works well, identifying the "correct features" requires domain- and problem-specific knowledge. We are trying to be very open about these limitations and challenges and at no point claim to select provably relevant features across all possible application domains. In particular, we have devoted section H of the appendix to discussing limitations of our method in detail. We apologise, but we do not understand the concern of the reviewer with our claims.

Can the authors to consider 3 points (1,0), (0,1), (-1,0), draw three disks centered at the points with a growing radius alpha and check that the resulting union of disks always has trivial 1-dimensional homology without cycles?

As discussed above, we have realised that we used a non-standard but a closely related definition of $\alpha$-complexes. In particular, our definition assigned every simplex a filtration value of its circumradius, as the definition in the literature does. However, the literature definition then updates the filtration value of some select "Gabriel" simplices below the top dimension to the filtration value of one of their co-faces. In order to avoid confusion, we now call our previous definition $\alpha^*$-filtrations, give a definition of $\alpha$-filtrations and discuss the relation between the two. In particular, both filtrations operate over the same set of simplices and assign the same filtration values to all top-dimensional simplices and all non-Gabriel simplices. We now formulate Theorem 4.1 for alpha^*-filtrations and give conditions for when the theorem applies to alpha-filtrations as well. We thank the reviewer for helping us to find this issue.

We will now try to explain why your situation does not pose a counterexample to the theorem. According to the definitions used in our paper (now called the $\alpha^*$-filtration), the filtration value of a $2$-simplex is its circumradius r. A fact from elementary geometry tells us that r is at least the maximum of $a/2$,$b/2$ and $c/2$, for $a$, $b$, $c$ the length of the edges of the simplex. $a/2$, $b/2$, and $c/2$ are the filtration value of the edges. Equality holds if and only if the points form a right-angled triangle, which happens iff two points lie on diametrically opposed points of the circumsphere of the simplex. When sampling from this circle, this case has probability mass $0$, which is why our theorem is formulated only with probability $1$. In all other cases, the $\alpha^*$-filtration value of the edges are smaller than the filtration value of the triangle, and a non-trivial persistent homology class forms in the $\alpha^*$-filtration.

Now let us consider the $\alpha$-filtration. For an acute triangle, there are no Gabriel edges and the claim from the $\alpha^*$-filtrations holds for the $\alpha$-filtration as well. For an obtuse triangle however, one of the edges is a Gabriel edge, meaning it gets assigned the (larger) filtration value of the $2$-simplex. We have updated the theorem to reflect our new definitions and naming scheme.

We want to emphasise that both the $\alpha$, as well as the $\alpha^*$-filtration assigns filtration values based on the circumradius of some $k$-simplex, i.e. the radius of the $(k-1)$-sphere defined by the $k+1$ vertices of the simplex. This is a different sphere than the spheres obtained drawing circles around the vertices of the simplex.

We have tried to very carefully explain our reasoning and the relevant definitions. In case you have further questions, we will be happy to answer them. As discussed above, from our point of view a main part of your review does not appear to relate to the central points and ideas of our paper. We would be very interested in hearing your opinion on this, especially on the apparent "major misunderstanding of persistence". Thank you in advance for your reply!

By node-level we mean assigning a feature vector to every single point, as discussed in the introduction, Figure 1 and the remaining sections. This is very different from trying to provide an isometry invariant of point clouds.

Is it correct that all your simplicial complexes, persistent diagrams, and feature vectors (assigned to points) are preserved under isometry (any distance-preserving transformation) of a given point cloud? If yes, then your outputs are isometry invariants.

Yes, these feature vectors are preserved under isometries, just as many other constructions like the clustering labels of any non-random clustering algorithm, some statistical descriptors, the number of points, or any function in the distances of the point cloud.

The class of "isometry invariants" of point clouds is thus a very diverse set of constructions with many different goals. When the literature talks about finding new isometry invariants of point clouds, they usually aim to find a single complex descriptor which completely distinguishes all non-isometric point clouds and is itself being preserved under isometries, ideally with low computational complexity. This is a very specific goal and very different from the goal of our paper. This is what we meant to express by the above quote, we apologise for not being specific enough.

In this case, it is a matter of scientific integrity at least to mention other isometry invariants of point clouds, especially because distance-based invariants are easier, faster, and stronger than persistence, also known for more than 20 years and studied at the point level under the names of local distribution of distances and local shape descriptors.

As we discussed above, there is a wide class of isometry invariants which try to solve very different poblems, often very different from the goals of our paper. We already benchmark against other "isometry-invariants" like diverse clustering algorithms and the local shape descriptors extracted by pgeof, and discuss various "isometry invariants" from TDA in the related work section. We will however add a reference to the suggested 2011 Memoli paper in the related work. Thank you for this reference!

Did you mean that this "wealth of literature" used complicated invariants (persistent homology) of unordered point clouds on atoms of protein chains that are naturally ordered along a protein backbone and hence have complete linear-size invariants?

To the best of our knowledge, yes. If the goal of the papers were simply to distinguish different proteins based on their associated point clouds, your objection would be valid. However for many more complex tasks, a better descriptor which is robust and organises the information in a good way is needed. It turns out that persistent homology is such a descriptor.

wealth of literature in TDA proves that these point clouds with topological structure appear in many applications, see our introduction for more references, or surveys like [1] or [2]

Have [1,2] explained that persistence is an isometry invariant of a point cloud? If not, other sources can also be helpful.

These surveys have not explicitly stated this. (However it is obvious from the fact that VR-persistence diagrams are functions in the distance matrix.) We suppose that this omission is due to the perceived relevance of this to the topic discussed in the surveys. What do you mean by stating "If not, other sources can also be helpful."?

Do you accept the counter-example on 3 points to the original statement for the classical alpha-filtration?

As discussed above, the theorem pre-revision was proven using the definitions given in the paper (and cited by you above), whereas the theorem in the revised version now holds for and is proved using the corrected definitions.

Because this substantial update took more than a week and the experimental results might be different for the newly proposed filtration,"reviewers reserve the right to ignore changes that are significant from the original scope of the paper", see https://iclr.cc/Conferences/2025/CallForPapers.

We do not know how to respond to this. We have never seen a ML conference rejecting a paper because the revision took a week. (In many ML conferences, the rebuttals will even only get published after a week, so should every paper be rejected then?) Furthermore, as we write in our rebuttal, we made a technical adaption to one of the definitions so that it reflects the definition used in implementations (including ours) and literature. This does not change any of the experiments. In particular, the scope of the paper remains virtually unchanged, and we have added only further evidence for our contributions based on feedback by the reviewers.

We thank the reviewer for their review and their feedback! We put a lot of effort into all of these points, and thus are very happy about the positive things the reviewer had to say about our introduction, the step-by-step description of section 3, our illustrations, the construction of TOPF and the set of experiments. We have uploaded a revised version of the paper in response to your and the other reviewers' feedback and will reply to your comments below.

Though acknowledged by the authors, the setting of Theorem 1 is very idealized. In particular, the -sphere has somewhat "maximal reach" and it is less clear to me that the theorem remain (with probability < 1 but still reasonable) valid when the underlying manifold is a "pinched" sphere with low reach. The theorem remains nice nonetheless and its validity in applications is not crucial to the work.

We agree with the reviewer that the setting of Theorem 1 is very idealised. Proving more general guarantees appears to be very hard due to the interplay of possible complex topological structures, noise, the non-linear nature of the underlying Delaunay triangulation of the alpha-complex, and the continuous nature of the harmonic representations.

Please note that another reviewer made us aware that our definition of $\alpha$-complexes had some technical differences to the definition of alpha complexes used in implementations and the literature. In order to avoid confusion and be open about this, we have renamed our previous definition to $\alpha^*$-complexes, and introduced the more commonly used definition as $\alpha$-complexes. We now state Theorem 4.1 for $\alpha^*$-complexes and give conditions on when it holds for $\alpha$-complexes as well.

Typo : "hull hull" in Theorem 1

Thank you for catching this typo, we have fixed it!

In the first experiment (Topological Point Cloud Clustering Benchmark), I believe that including topological regularization term in point embedding methods (node2vec and pointnet) has been attempted in some works (e.g. in Topological Autoencoder of Moor et al., some experiments in Optimizing Persistent Homology based function of Carriere et al., Topological Node2vec of Meehan et al., etc.). I believe that this would not invalidate the claims made in that paragraph, namely that these methods requires specific training to work and may be slower than the proposed approach, but it may be worth to briefly mention these.

Thank you for these related references! We will include a discussion of them in a future work! In Topological autoencoders, the authors basically set up a topological loss functions that ensures that topological features of the original point cloud are preserved in the latent space of the autoencoder. This is different from the approach TOPF takes, as TOPF tries to detect and extract topological features. Similar things can be said about Topological Node2Vec and some experiments in "Optimizing Persistent Homology Based Function", or some experiments of the newer "Diffeomorphic interpolation for efficient persistence-based topological optimization" by Carriere, Theveneau, and Lacombe: They all try to come up with ways to preserve global topological structure while embedding a point cloud, while we on the other hand construct the embedding based on the relationship between the points and the global topological structure.

We will add a discussion of this to the paper and thank you for the suggestion!

1. [..] Is there a way to somewhat "smooth" the choice of to avoid relying on a hard threshold but rather considering (possibly infinitely) many dimensions but whose relative importance would be weighted by $\ell_i$? I ask this question out of curiosity, I do not expect an extensive answer in the rebuttal.

We thank the reviewer for this excellent question! You are correct that in certain circumstances the number of features picked by the heuristics is not stable. In general, it seems impossible to design a heuristic that is stable in this regard without any prior knowledge on the data. However, the current implementation of TOPF already permits to extract a fixed number $N_d$ of topological features per dimension for reasons outlined in your question. When then weighing the extracted features by a suitable weight function in the length of their associated persistence bar (like exponential/polynomial functions of high enough d), this feature will converge for large $N_d$. It is very plausible that this is the desirable approach in some applications, especially those requiring fixed-dimensional features/positional encodings. Choosing the correct weight function seems like an interesting research question for future work. We have added a discussion of this to appendix F and consider adding a brief discussion to the main text.

In case you have further ideas on this, we would be very happy to hear them!

Thank you again for your feedback! We are happy to answer any follow-up questions.

Thank you very much for your detailed review and the many good questions and suggestions! In particular, we are of course very happy that you commended liked quite a few things about our paper. :) We have uploaded a revised version of the paper based on the feedback, and will reply to your points below

Algorithm 1 is too schematic and it is hard to reproduce the method based on it. It can heavily benefit with more detailed substeps;

Thank you for this suggestion. We will add details and substeps for Algorithm 1 and thus try to make it easier to reproduce the algorithm. Because TOPF and the involved concepts are rather complicated, we are afraid that a complete reproduction of the algorithm is not possible without reference to the explanations of section 3 or the submitted code.

How sampling density affects performance of the method? For example, what happens with predicted labels in Figure 8 if we reduce sample sizes of point clouds by 2x? 3x? 5x? My concern is that the model might not work well on sparse point clouds.

Thank you very much for this valid and valuable remark! In the selection and construction of the point clouds of the TCBS, we have already tried to incorporate very different sample sizes, from very sparse point clouds with 158 points (Ellipses) to 4600 points (2Spheres2Circles). Based on your comment, we have now introduced an experiment (See Figure 15) where we decrease the sampling density for our point cloud with the highest original sampling density (2Sphers2Circles). It shows that TOPF still maintains an ARI over 0.9 even for 700 points, which is a more than six-fold decrease in sampling density, and still performs reasonably well for even stronger downsamplings. We believe that the robustness from TDA thus allows TOPF to perform well on sparse sets, especially when compared with modern Computer Vision techniques using neural architectures.

Similarly, what happens with the method if we have non-uniform sampling from the manifold? For example, what happens when the target manifold has a high-curvature area that is not sampled densely enough?

Thank you again for this comment! Already in the TCBS we have included datasets with samling irrgularities: In Ellipses, the small 2 ellipses are sampled much more densely than the outer ellipses, in Spheres+Grid, the middle grid is sampled much more sparsely than the remaining blobs, and finally in Spaceship, the two small ellipsoids which look like ears are sampled much more densely than the remaining ellipsoids.

The quantitative experiments seem to indicate that TOPF can adapt well to these cases, at least if every topological feature has a homogenous enough sampling density/noise level. The reason for this is that by making use of the birth and death time of the associated bars in the persistent homology diagram, TOPF can pick a scale/alpha radius for each of the features independently. (See our answer to the next point.)

My understanding is that the epsilon radius of local Delaunay triangulation (Definition B.2.) is the hyperparameter of the method. It is not clear how this parameter should be chosen and it looks like it can strongly affect the results: if it is too large, local Delaunay triangulation will start closing holes; if it is too small, the manifold will be disconnected. Do authors have some practical suggestions how this radius should be chosen, especially if we don’t have good priors about the data?

The epsilon radius of Delaunay triangulation is not a hyperparameter of the method and quite a nice trick (we think.) All hyperparameters are listed in appendix D "Hyperparameters". Every topological feature corresponds to a bar with a death time $d$ and a birth time $b$ in the persistent homology diagram. Thus, for each feature we have a range of possible alpha-radii where the feature is guaranteed to exist. Using the interpolation-parameter $lambda$ as hyperparameter with default $\lambda = 0.3$, we then choose as $\alpha$-radius $r=b^{1-\lambda}d^\lambda$, the weighted geometric mean. In dimension $0$, all birth times are $0$, hence we have to pick the weighted arithmetic mean. We do a sensitivity study on lambda in the appendix and discuss this procedure in section 3 step 3. **In conclusion, TOPF makes use of persistent homology to choose a provably valid $\alpha$-radius for every feature.$$

Theorem 4.1 has very limited practical applicability to actual 3D data analysis. It assumes two sets of points: one set is sampled from the unit sphere; another set is sampled with distance at least 2 to this unit sphere. For the majority of 3D point cloud applications, point clouds are normalized to unit cube. My understanding is that in this case assumptions of Theorem 4.1 do not hold. Is my understanding correct? Maybe the setting that I have described can be adapted for Theorem 4.1 somehow?

The goal of theorem 4.1 is to prove that TOPF works in a very specific and constructed setting. This is to show that the theory can work in principle. You are, however, right that Theorem 4.1 does not give widely applicable guarantees in real-world 3D data analysis. Proving such guarantees appears to be very hard due to the interplay of possible complex topological structures, noise, the non-linear nature of the underlying Delaunay triangulation of the alpha-complex, and the continuous nature of the harmonic representations. (Of course, we can adapt the argument to scaled versions of the hyper-sphere.)

Please note that another reviewer made us aware that our definition of alpha-complexes had some technical differences to the definition of alpha complexes used in implementations and the literature. In order to avoid confusion and be open about this, we have renamed our previous definition to $\alpha^*$-complexes, and introduced the more commonly used definition as $\alpha$-complexes. We now state Theorem 4.1 for $\alpha^*$-complexes and give conditions on when it holds for $\alpha$-complexes as well.

On a related note, what is the normalization of the data for the experiments? Are point clouds normalized to unit cube? Is this normalization the same across all evaluated methods?

We did not normalise the point cloud, as we assumed some underlying meaning in the scaling of the data.

PointNet baseline is very weak. It was introduced in 2016 and was superseded by a large number of architectures: PointNet++, several versions of PointTransformers and other attention-based architectures. PointNet performance on the majority of benchmarks is significantly worse than for all of the mentioned methods. More up-to-date method can give significantly better performance on proposed benchmark.

We agree with the reviewer that PointNet is not the current SOTA on ShapenetPart Segmentation. However, PointNet is still in the range of 6% IoU of the current SOTA, in comparison to the 42% difference on TCBS with TOPF. This suggests that the performance of PointNet is still roughly representative of other similar Deep Learning approaches.

Perhaps more interesting than benchmarking against PointNet is benchmarking TOPF against other modern unsupervised (or weakly supervised) learning methods for point cloud feature extraction. We have done this with WSDesc[1] (pretrained on 3DMatch) on suggestion of another reviewer. WSDesc is competitive on current benchmark tasks. While WSDesc has a better performance than PointNet on the 3D datasets, TOPF still outperforms it by a wide margin on the more challenging sets of the TCBS. In addition to the 32-dimensional features produced by WSDesc, we have added a feature channel for the normalised 3d coordinates, because this greatly improved the performance of WSDesc on TCBS.

The obtained results are very interesting: On the easiest 3D dataset, 2Spheres2Circles (with a large (>4000) number of points and uniform sampling), WSDesc matches the performance of TOPF with an ARI of 0.95. On Spaceship, WSDesc achieves an ARI 0.76 (TOPF: 0.92). Looking at the results directly, we interpret this as WSDesc struggling with the non-uniform sampling across the different features. Finally, on SphereinCircle, WSDesc achieves an ARI of 0.46. This is better than most other non-toplogical methods, but significantly worse than TOPF with 0.97. In our interpretation, this has to do with the sparseness of just 267 points.

On the 2D datasets (which we embedded into 3D) WSDesc performs very poorly, which might be expected because it was trained for 3D datasets. This shows that on the TCBS, TOPF is competitive even against weakly supervised methods which excel on current benchmarks.

[1] Li, Lei, Hongbo Fu, and Maks Ovsjanikov. "WSDesc: Weakly supervised 3D local descriptor learning for point cloud registration." IEEE Transactions on Visualization and Computer Graphics 29.7 (2022): 3368-3379.

Dear reviewer,

thank you for your reply and your remarks. We apologise for the late reply!

We have already incorporated a large number of your points into the revised pdf which is uploaded to openreview. In particular, we have done the following:

I recommend removing Theorem 4.1 completely and moving Figures 13 and 15 to main paper.

We have done this and briefly discuss these experiments in the main paper. Thank you for your suggestion!

Experimental evaluation should use 'native' normalizations for all baselines.

We have re-run WSDesc with and without normalisation to the unit cube on all of the datasets of TCBS with 50 tries and looked at the results in detail. As reported, this does not change the mean ARI achieved by much, and in particular it worsens it on most of the datasets.

This shows that even WSDesc can utilise the natural scaling in our input point clouds. We will post exact numbers on this before the end of the discussion phase, our server is not in a good mood at the moment and we are currently trying to talk to it, we apologise.

UPDATE: We apologise, but our server is still experiencing issues and we cannot post the results here. We will, however, include them in the final version of the paper.

I strongly recommend including results on sampling regularity and sampling density (for baselines as well).

Thank you for the feedback, we now report the desired results in the uploaded pdf in Fgure 15 and 16.

I have also checked authors' discussion with reviewer smnw and I agree with the following reviewer's points:

Thank you very much for reading the discussion with other reviewers! We value your time and a second opinion highly!

alpha*-complexes might be confusing both for readers who are familiar and who are unfamiliar with the theory;

Thank you for this point! We have moved Theorem 1 to the appendix, and now only talk about $\alpha^*$-complexes in the theory section of the appendix, where we have added a detailed explanation on the role both of these concepts play. We believe that this reduces the confusion, in particular because we have now fixed the previously misleading definitions, as remarked by reviewer smnw.

proposed work seem to be proposing isometric invariant and thus local shape descriptors probably should be mentioned in related work.

This is a point where we agree with the review, we have already added a discussion of the 2011 Memoli paper to the related work section in the appendix. Thank you for pointing this out again: we are open to more suggestions.

We will include a detailed pseudocode version and an even further expanded related work section in the camera-ready or resubmitted version of the paper.

We want to sincerely thank you for all the time and effort you invested in reviewing this paper and for all the constructive feedback we have received! Despite all the negative publicity regarding the state of the peer review process at large ML conferences, this was great!

We believe we have significantly improved our paper since your first review and were able to address the majority of your concerns. We would thus be interested to hear how your assessment of the paper has changed.

Thank you again! The authors

We thank the reviewer for their review and their detailed feedback! We have uploaded a revised version of the paper based on the feedback, and will reply to your points below

a) The Topological Clustering Benchmark Suite => TPCC is introduced by the authors in this paper to evaluate the effectiveness of the proposed clustering method. The construction of these datasets in this benchmark has not been sufficiently discussed in the paper/appendix. For instance, it is unclear how the ground truth labels have been obtained since all the datasets are synthetic in nature.

We constructed the benchmark by sampling from topologically different shapes with varying sampling density in different ambient dimensions. For example for the point cloud 2Spheres2Circles, we combined points sampled from two spheres and two circles. We then divided the point cloud into four ground truth clusters, one for each of the two spheres and two circles and assigned every point the cluster corresponding to the object it was sampled from. Thus, the ground truth label of a point corresponds to the topological structure/object it was sampled from.

We have now added an explanation for this to appendix C. Thank you very much for your comment!

b) 3D datasets => What are the ground truth labels for these datasets, and how have these labels been obtained?

We only have ground truth labels for the 3D datasets of the TCBS, which we obtained as described above. In a reply below, we will explain the different datasets of the paper.

a) There are a total of 12 datasets presented in the paper: seven 2D datasets in the TPCC benchmark and five 3D datasets, as shown in Figure 5. Then why are results on only four 2D and three 3D datasets presented in Table 1?

There seems to be some confusion over the data sets used in the paper. We are grateful for the reviewer to raise these issues. We have subsequently made the passages in the paper more clear and will try to explain our data sets in the rebuttal: There are in total 11 (now 12 in the revised version) data sets used in the paper. 7 of these are synthetic and are contained in the topological clustering benchmark suite (TCBS) and thus have ground truth clustering labels, obtained as described above. Out of these 7, there are four 2d and three 3d data sets, as described in the caption of Table 1, in section C in the appendix and in Figures 7&8. This is how we obtained the ground truth labels for the 3D data sets for the experiments reported in Table 1.

The remaining point clouds in figure 4 (a), b), c) and e)) are protein coordinates, or the state space of a simulation of a Lorentz attractor, or the latent space of a VAE. Furthermore, we have now added a high-dimensional space of points sampled from the conformation space of cyclooctane (Fig. 11). We have not used ground truth data for these four point clouds, as it does not seem to be clear what would constitute a ground truth. We could have of course came up with a notion of ground truth aligning with the results of TOPF, but we felt this to be disingenuous. Thus we have not included benchmark results on these point clouds in Table 1. We note however, that qualitative evaluation in Figure seems to indicate that TOPF is able to recover meaningful topological features.

We have now explained the construction in more detail in the appendix and thank the reviewer for their feedback!

Why were auxiliary points added in Cys123?

In the experiment with Cys123, we wanted to show that TOPF can detect so-called protein pockets. Because pockets do not form holes detectable by TDA straightforward, the literature suggests adding (and later removing) these points on the convex hull, which turns the pockets into holes, see for example "Novel definition and quantitative analysis of branch structure with topological data analysis." Haruhisa Oda et al., 2024. This demonstrates just another possible use-case of TOPF. We have added a more detailed explanation of this to the paper to appendix E and thank the reviewer for their feedback.

Please provide a quantitative comparison with the baselines on the 3D datasets.

As we discussed above and mentioned in the caption of table 1, three of the datasets, namely 2Spheres2Circles, SphereinCircle and Spaceship, are 3D datasets. There does not seem to be a major performance difference of TOPF on 2D and 3D datasets.

Why are some of the run-times in Table 1 0.0 seconds?

This is due to rounding. Times less than 0.05 seconds were rounded to 0.0. We have their entries to <0.1 to avoid confusion.

What is the breakdown of TOPF run-times in terms of Steps 1-4 of Algorithm 1?

We have now added a detailed breakdown of the individual steps of TOPF on the point clouds of TCBS to Table 2. In summary, computing the persistent homology and the harmonic projections constitute the majority of the runtime. Thank you for the suggestion!

3. Presentation: The algorithm presentation needs significant improvement. Steps 3 and 4 of the TOPF algorithm are hard to read.

We are sorry to hear that steps 3 and 4 of the TOPF algorithm in section 3 are hard to read. We have tried very hard to make the entire paper as accessible and easily understandable as possible. Section 3 is without doubt the most technical section, where an understanding relies on a good grasp of concepts of algebraic topology and discrete Hodge Theory. We have thus tried to write the paper in such a form that the remaining sections are accessible with only a very rough understanding of the technicalities of section 3. Nonetheless, we believe they are important to include in the paper to enable exact reproduction of our algorithm. However, we have re-read the section and will try to improve readability by adding explanations and being more clear about the concepts used in the revision period.

Please explain these steps using a concrete, small-scale toy point cloud dataset.

As referenced in the introduction of section 3, we provide an overview with a toy point cloud in Figure 1. Please note that the steps from Figure correspond to the steps in Section 3. Furthermore, we have illustrated the algorithm in the three subfigures of Figure 3. We believe that these two figures are good illustrations, but are open to feedback by the reviewer.

The terms “simplex-valued harmonic representatives” (line 337) and “simplex-valued vector” (line 341) are used without any explanation.

We have rewritten the parts and added an explanation. Thank you for raising this issue!

What is the role of interpolation parameter ?

The mathematical role of the interpolation parameter $\gamma$ is explained right after its introduction in Step 3. Given a topological feature with birth time $b$ and death time $t$, $\gamma$ controls the time step $t$ where we construct the associated simplicial complex with $t=b^{1-\gamma}d^\gamma$. (For homology dimensions above $0$. In dimension $0$, the birt time of the feature is $0$ and thus we use linear instead of exponential interpolation.) A more intuitive explanation of the interpolation parameter can be found in section D.1 'Hyperparameters' of the appendix. In a nutshell, larger interpolation parameter will result in smoother topological features, whereas a smaller interpolation will lead to 'sharp' features. We just noticed that the subsections D.1 'Hyperparameters' and D.2 'Runtime' were partially swapped, we have corrected this issue.

4. Novelty: Which component of the algorithm is novel? Is it step 4, since Steps 1-3 are standard tools of TDA?

First we would like to remark that novelty is often not only a sum of the individual components, but much more the entire idea to combine previous techniques in a new way to obtain a novel result/goal, see for example Michael Black's Guide to novelty.

The entire idea to compute node-level features relating the global topology in terms of persistent homology back to the local points is very novel, the only somewhat similar papers being [1] and [2], although we highlight the differences and the advantages of our approach in the related work.

Step 1, computing persistent homology is indeed a standard tool of TDA. Step 2, picking significant generators, was done in [3], but we have introduced significant refinements to the heuristic, see appendix F "How to pick the most relevant topological features", and furthermore realised that for our applications, considering homology instead of cohomology is the right approach. Step 3, Projecting homology generators to the harmonic space was recently considered in [1] and [4], however, there the projections where performed at the birth of the persistence bar. In comparison, we have introduced the interpolation parameter, leading to smoother and more stable topological features. Furthermore, we have introduced a simplex weighting improving the harmonic representatives, and present an easy-to-solve least squares problem to compute them using the Hodge decomposition, which we are unaware of previous applications. (Edit: There are prior similar implementations, phrased differently) Step 4 is novel in this context in this form.

We hope to have answered the question of the reviewer!

[1] Gurnari, Davide, et al. "Probing omics data via harmonic persistent homology." arXiv preprint arXiv:2311.06357 (2023) [2] Grande, Vincent P., and Michael T. Schaub. "Topological point cloud clustering." Proceedings of the 40th International Conference on Machine Learning. 2023. [3] De Silva, Vin, and Mikael Vejdemo-Johansson. "Persistent cohomology and circular coordinates." Proceedings of the twenty-fifth annual symposium on Computational geometry. 2009. [4] Basu, Saugata, and Nathanael Cox. "Harmonic persistent homology." SIAM Journal on Applied Algebra and Geometry 8.1 (2024): 189-224.

5. Limited Applicability: TOPF only obtains meaningful results on point cloud with topological structure. However, many datasets do not have such a structure.

It is true that many point clouds do not have a topological structure. However, the research on Topological Data Analysis over the last decade reveals that there are a wide range of applications where point clouds possess topological structure, see for example the wide range of applications cited in our introduction from biology, physics and computer science. We thus believe that these topologically structured point clouds are important enough to warrant research. In particular, as our experiments show, TOPF clearly outperforms any other method in this niche of topologically structured data.

Furthermore, TOPF is computationally expensive on datasets with a large number of points (e.g., those obtained from CAD designs or 3D Scanners). While the authors mentioned using landmarks to improve performance, no experiments in the paper did so. It seems the method is only practical in a very limited setting, which is points with only topological shapes and a small number of points.

In Figure 10 in the appendix, we provide experiments comparing the runtime of TOPF on point clouds with and without sampling enabled with close to 50000 points. We believe that most point clouds can be sampled down to 50.000 points. For larger point clouds, the Ripserer.jl library used in our implementation sometimes crashes, which is the only thing prohibiting experiments without downsampling on larger point clouds. In Figure 15, we have added an experiment on how downsampling affects performance of TOPF.

While we agree that TOPF does not have the performance of some Computer Vision models running on multiple GPUs, we have shown that using the base implementation in conjunction with landmark and random sampling can make TOPF scale to rather large point clouds.

1. What is the maximum homology dimension in the experiments?

For the experiments in the TCBS, we set the persistent homology dimension to the maximum possible dimension permitted by the ambient dimension, which was 1 and 2 respectively. For the experiments with real data, we use prior knowledge on the topological structure we were looking for to set the homology dimensions: For the NALCN channelosome and GroEL proteins, we wanted to detect the loops/1-dimensional holes in the protein structure, and looked at dimension 1. For the Cys123 (with the added convex hull), we looked for protein pockets and thus only looked at dimension 2. For the VAE latent space embedding experiment, we looked for loops informed by the sampling strategy, and thus set the dimension to 1. Finally, for the Lorentz attractor data set, we again looked for loops around the non-attractive fixed points and thus used a homology dimension of 1. We assume that in many use cases, we will already** know for what kind of topological features** we will be looking for. However, we can always run TOPF with a larger maximum homology dimension and TOPF auto-selects the relevant dimensions by comparing the life times of features across dimensions. Which is of course more prone for error than using prior knowledge.

What is the relation between the output of step 3 of Algorithm 1 with the Circular coordinates of De Silva & Vejdemo-Johansson (2009)?

We discuss differences in our step 3 with previous literature including the circular coordinates in an above comment. The authors of the cited paper integrate the harmonic cohomology representative to obtain, for every node, a circle-valued coordinate on a circle. This assigns the coordinate to the point independent of whether the point lies close to the circle the coordinate is representing. (This is because by Brown's representability theorem, every cohomology class is representable by a map of the entire space into an Eilenberg-MacLane Space, which is for dimension 1 a circle.) These coordinates then don't measure how much a point belongs to a topological features (as we do) but rather at which point of the circle (even if it does not lie on the circle) the point lies. This then only works for homology of dimension 1. This is a very interesting and somewhat orthogonal approach, but for example would not work for clustering on the TCBS. (We apologise for the mathematical explanation but hope that this helped to at least gain some intuition.)

We want to again thank you for your feedback, remarks, and questions. We believe that these already resulted in a better paper. We would be happy to answer any follow-up questions, clear out any remaining misunderstandings, or hear whether your assessment of the paper has changed. In particular, we are looking forward to the discussion!

Thank you very much for your reply! We are happy that we were able to resolve your concerns regarding the dataset, novelty, and experiments! We will reply to your comments below:

My question about computational complexity remains unanswered

Thank you for your reminder! We apologise, we forget to address this point in our previous reply. We will discuss the complexity of the involved steps:

1a). Constructing the complex: The Vietoris--Rips complex will have $O(n^{k+1})$ simplices, where $n$ is the number of points and $k$ the maximum homology dimension. Apart from this, there is no significant computational effort needed. Note that we can decrease this if we set a maximum VR-radius, as done in many works from the literature. Computing the alpha-complex in ambient dimension $d$ is equivalent to computing the convex hull in ambient dimension $d+1$, which has a complexity (edit:) and number of simplices of at most $\sim O(n^{(d+1)/2})$.

1b). Computing persistent homology and generators: As we need the homology generators, we use the recently introduced involuted persistent homology [1]. While the authors discuss the runtime performance, they do not give a concrete complexity bound in [1]. However, they deduce that it has a similar run-time to usual persistent cohomology computations, whose performance is for example discussed in [2].

2. Picking the significant generators has negligible impact.

3. Computing harmonic projections: Given a homology representative $r$ in dimension $k$, computing the harmonic representative amounts to computing a sparse matrix vector product of $B_{k}x$ and a sparse least squares problem lsmr(B_{k},r), i.e. solving $\min_{x\in \mathbb{R}^{S_{k+1}}} \lVert r-B_{k}x\rVert_2$ where $S_{k+1}$ is the number of $(k+1)$-simplices in the simplicial complex and $B_{k}$ has $(k+2)S_{k+1}$ non-zero entries. This is a sparse least-squares problem, which we solve using the iterative sparse solver lsmr [3]. Because this is an iterative solver, the authors do not give runtime complexity, but discuss the computational requirements in [3].

4. Pooling and averaging over the simplicial neighbours has negligible runtime constraints.

We have added the above discussion to the section on runtime in the appendix (which we will upload after finishing these comments) and thank the reviewer for suggestion!

[1] Čufar, Matija, and Žiga Virk. "Fast computation of persistent homology representatives with involuted persistent homology." Foundations of Data Science 5.4 (2023): 466-479.

[2] de Silva, Vin, Dmitriy Morozov, and Mikael Vejdemo-Johansson. "Dualities in persistent (co) homology." Inverse Problems 27.12 (2011).

[3] Fong, David Chin-Lung, and Michael Saunders. "LSMR: An iterative algorithm for sparse least-squares problems." SIAM Journal on Scientific Computing 33.5 (2011): 2950-2971.

Why $\alpha$-complex is built in both Ripserer and gudhi, twice?

This is just an implementational detail that will be optimised in a future release of TOPF. The implementation of above cited method [1] is in Julia, whereas we wanted to provide a python package while minimising the needed communication between julia and python.

What is “Julia output”?

This is the communication time between Julia and python. We did not want to rely on the sometime complicated setup of PyJulia. Again, this is an implementational detail that does not have an impact on the theoretical contributions of the paper.

While we agree that TOPF does not have the performance of some Computer Vision models running on multiple GPUs, we have shown that using the base implementation in conjunction with landmark and random sampling can make TOPF scale to rather large point clouds.

Are you referring to Figure 11? What do you mean by “mixture of landmark and random sampling” in the line: “To counteract the increase in computational complexity, TOPF automatically downsamples the point cloud using a mixture of landmark and random sampling.”?

We believe we are referring to Figure 10 "Runtime on two examples from Topological Clustering Benchmark Suite while increasing point density.", but the ordering could have changed.

By a mixture of landmark and random sampling we mean that in the first step, we sample down the point cloud randomly, and afterwards select points by iteratively adding the point with the largest distance to the already selected points.