Point-Level Topological Representation Learning on Point Clouds

Thank you for your review and your valuable feedback!

The experiments and the visualization are good. I wonder whether it is possible to evaluate on more diverse tasks, like ModelNet40 classification, ShapeNet segmentation, and S3DIS segmentation, like many point cloud papers evaluate.

Thank you for your comment! The point clouds in datasets like ModelNet40 or ShapeNet have none to very few topological features, while the neural architectures learn to extract some different form of features. However, TOPF was specifically design to extract the topological features from homology. It might be interesting to extend the ideas from TOPF to geometrical information as well in future work, but this will require many new ideas and is out of scope for the current paper. The above is the reason why we have introduced the novel Topological Clustering Benchmark Suite to benchmark TOPF.

Thank you very much for your careful review and feedback!

Weakness: It could be better to analyze the runtime of the computational complexity of persistent homology regarding different point cloud sizes.

Thank you for this suggestion! We analysed the computational complexity in appendix E.2 in detail, broken up into the different steps of the TOPF pipeline. Furthermore, we provided numerical experiments for runtime while increasing the size of the point cloud up to 40,000 points in Figure 11. Finally, we provide the number of points in the point clouds of the Topological Clustering Benchmark Set in Figure 8 in the appendix and provide the mean runtime of TOPF in Table 1.

We hope to have addressed your concerns and want to thank you once again for your review!

Thank you very much for your detailed and thorough review. We will reply to the raised issues in as much detail as the character limit permits.

Claims And Evidence: Thank you for your detailed feedback! While we still believe that the listed contributions are technically true, we now realise that parts might be misleading. We omit the “provably” from (ii). Regarding point (iii), we interpret this differently: When downscaling and using heterogeneous downsampling, from a certain degree onwards, there is no topological signal left in the data. The other algorithms detect simpler structural features and thus are not affected. However, without topological features, the performance of TOPF degrades, which is some form of sanity check. In Figure 16, Bottom left, we see that even a downsampling factor of 0.2 significantly deteriorates the topological signal.

However, we will write "[...] robust to moderate noise [...]".

I was wondering what performance classic clustering algorithms based on manifold distances would achieve on the datasets,

This is a good suggestion: PAM using Isomap distances has a mean ARI of 0.58 with 0.39/0.56/0.78/0.13/0.63/0.58/0.94 on the individual datasets. This puts PAM+ISOMAP between the performance of clustering algorithms directly on the points (kMeans mean ARI: 0.44) and TOPF (mean ARI: 0.86). We will add this to table 1.

a short description of TPCC should be added to the main part together with a list of changes done for TOPF and their effects.

Thank you for this good suggestion, we will do this!

Comments and Suggestions: Thank you very much, we have addressed your comments.

Questions for authors:

[...] Would you elaborate on how you identify homology and the kernel of the Laplacian and how this relates to Step 3 in section 3? [...]

This is a good question! Formally, the cochain space is the dual of the chain space. For finite-dimensional vector spaces with the standard inner product, there is a canonical isomorphism between the two spaces induced by sending a vector $v$ to the linear extension of the map sending $v$ to $1$. Thus, we can identify the chains and cochains. We have the boundary maps $B_i$ and the coboundary maps $B_i^T$. By the univ. coefficient theorem real-valued homology and cohomology are isomorphic, with $H_i=\ker B_i/Im B_{i+1}\cong H^i=\ker B^T_{i+1}/Im B_i^T$. Thus, using the canonical isom. to the dual vector space, $ker B_i$ are homology and $ker B^T_{i+1}$ are cohomology representatives. $\ker L_i$ is the intersection of $\ker B_i$ and $\ker B^T_{i+1}$. Hence every element in $\ker L_i$ automatically is a homology representative. This gives us an explicit isomorphism between $\ker L_i$ and the (co-)homology.

In step 3, we start with some arbitrary homology representative r in $\ker B_i$. However, we want a harmonic representative h in $\ker L_i = ker B_i\cap ker B^T_{i+1}$ for the same homology class, i.e. with h =r - c for c in $im B_{i+1}$, the curl space quotiened out in homology. By the orthogonality of the Hodge decomposition, this c is given by the projection of r to the curl space $im B_{i+1}$. We will discuss this in step 3 of section 3 and in more detail in appendix B.2!

I am specifically curious why TOPF is able to outperform TPCC on some datasets (HalfCircles, Ellipses, ...) by a large extent but achieves similar performance on SphereInCircle .

As you mentioned, we already give a brief comparison. The performance differences of TPCC and TOPF on different comes mainly down to how “difficult” and “robust” the topological structure encoded in the datasets is. In particular, 2Spheres2Circles and SphereInCircle are directly sampled from unions of manifolds without any noise and sufficient sampling density. In comparison, in Ellipses and Spaceships, the topological features have shorter life times and live at different scales. There is no single scale containing all features, and for most holes/voids, there is not even a scale whithout noisy holes with small persistence present in the filtration. TOPF can deal with these situations, whereas TPCC cannot. For the difference on HalvedCircle, we posit that this is due to the better feature aggregation of TOPF. TPCC requires to perform subspace clustering on an harmonic edge embedding, which is both unstable and sensitive to parameters, which we think TPCC has no information to choose correctly in this setting.

Do you have an explanation or conjecture why TOPF performs rather weak on the 4Circles+Grid data?

Figure 8&9 show the ground truth and TOPF clustering on 4Circles+Grid. In short, TOPF chooses suboptimal scales which are not well-enough connected. Figure 10 supports this interpretation, increasing the interpolation hyperparameter lambda improves TOPF performance on this dataset. We have used a fixed set of hyperparameters for all experiments on the TCBS for transparency. We will add an extended discussion of this to the paper.

Thank you very much for your feedback! We are happy about the many strengths of TOPF identified by you. We will now address your comments:

Weakness 1: Although the author analyzed TOPF from a theoretical perspective, I still think that the author's TDA cannot be actually applied in the current field of point cloud analysis, and it is currently only in the theoretical analysis stage. Its high complexity and time consumption make it difficult to apply in practice.

Thank you for your comment! We applied TOPF on synthetic and a variety of real-world datasets and conducted experiments on large-scale point clouds in appendix E. Furthermore, persistent homology, which has similar runtimes, has been used successfully in the literature and many applications. We believe that this is convincing evidence that TOPF is a valuable scientific contribution. Because TOPF has the single goal of extracting topological information from homology, we do not claim it to be relevant in all point cloud tasks.

(Indexing continues at weakness 4 in your review.)

4. Please refer to our rebuttal to reviewer yYZ5. Furthermore, we evaluate large-scale scenarios of point clouds with up to 40,000 points in E.2, showing that TOPF, particularly with the proposed landmark downsampling heuristic, is feasible even on large point clouds.

5. This method may not be applicable to very high dimensional point clouds,

Thank you for your comment! TOPF works well on high-dimensional data sets. In the paragraph “Embedding Space of Variational Autoencoders and High-dimensional spaces” of sec. 4, we apply TOPF successfully to high-dimensional latent spaces and directly to an 8478(!)-dimensional image space. In Figure 4, we show a projection of the results of TOPF on a 24-dimensional point cloud.

and the authors should explicitly discuss its limitations.

We agree with your comment: We already give a summary of the limitations of TOPF in sec. 5. We discuss the limitations in great detail in section I: Limitations in the appendix.

6. Is it possible to provide a detailed computational complexity analysis of TOPF?

Thank you for your comment! We discuss the computational complexity in detail in section E.2 in the appendix, split into the different steps of the algorithm and conduct experiments regarding the scaling behaviour.

7. Compared with existing neural network-based methods, can TOPF outperform them? Please provide relevant experiments and analysis.

Thank you for your comment! TOPF outperforms neural network architectures on the task TOPF was designed for, namely extracting topological features from point clouds. We benchmark TOPF’s performance on the TCBS against the neural network-based methods PointNet and WSDesc, see Table 1. Furthermore, we have now added a benchmark against DGCNN (Wang et al.). TOPF outperforms these approaches (pretrained on large-scale part segmentation datasets) with a mean ARI of TOPF of 0.86 to 0.44 (PointNet) 0.39 (WSDesc) 0.64 (DGCNN). We discuss this in more detail in Section 4. We have pretrained the neural architectures on large-scale shape segmentation data sets. For a detailed experimental setup, see Appendix F: More Details on the experiments.

The legend of Figure 1 is too long, and it is recommended to split it into multiple sentences.

Thank you for this suggestion! We hope the caption of Figure 1 to be self-contained and an explanation for the figure, thus we do not know how to further shorten it. Do you have any suggestions? We already broke down the caption into 10 short to medium-length sentences. Does this suffice for you?

The resolution of Figure 3 needs to be improved to enhance readability.

We are apologise, but we do not understand this comment. On our end, the figure has sufficient resolution and an associated file size of 1.5 MB. Could you further elaborate on your suggestion?

Thank you again for your review! Based on your feedback, we believe the paper is in an even stronger state than before! We believe to have addressed a majority of your concerns and would be interested in hearing back from you!