Paper: Topo-Diffusion: Topological Diffusion Model for Image and Point Cloud Generation (ICLR-2024)

Reviewer comment (Novelty):
The related works section of diffusion models is missing many key works such as [1,2].

Author response:
Thank you very much for your comments! We have included [1] [2] and some other existing works in our paper. We have rewritten the introduction and related work sections with a new perspective, in which we mainly focus on the underlying diffusion mechanisms.

Reviewer comment (Presentation):
Figure 2/3 should have its axes labeled for greater clarity.

Author response:
Thank you for your suggestion. We have redrawn Figures 2 and 3.

Reviewer comment (Theory):
Even though TDA is very powerful in characterizing the structural topological information, either local or global, it has limitations in the detailed description of local geometric features. It seems that the current model is more suitable for data with strong topological properties but may fall short when the detailed geometric information also plays an important role.

Author response:
**Response to weakness:** Thank you very much for pointing out this concern. We have conducted additional experiments on three real datasets, which are labeled faces in the wild (LFW) dataset, the circuit reconstruction from electron microscopy images (CREMI) dataset, and NASA satellite image dataset (NASA’s Satellite Imagery) to evaluate our proposed method (which would further strengthen our work and facilitate readers' better understanding of the applications of our Topo-Diffusion). We will conduct and provide more experimental results in the final version. In the revision, we provide results on human face dataset to demonstrate the performance of our Topo-Diffusion to enhance the image quality. Focusing on structure information,  we have also conducted additional  experiments on cell image and satellite image datasets and shown how the structural information in synthetic image can be captured by Topo-Diffusion.  

We have listed the following quantitative results to demonstrate the performance of the proposed Topo-Diffusion model. We have provided qualitative results of generated images from CREMI dataset *(see Figure 19 in Appendix)* and the NASA dataset *(see Figure 18 in Appendix)* to demonstrate the effectiveness of including persistence homology information in the Diffusion-based model. The qualitative results show that the persistence homological features enhances the generative model to coverage much richer topological information. The diffusion model can improve the diversity of synthetic data. Topo-Diffusion can improve the quality of generated images in the aspect of capturing topological/geometrical information as well as generating high-quality images. 

**Table 1: Performance on the CREMI dataset.**
|                      |     WGAN   |  Topo-GAN   |   DDPM    |  Topo-Diffusion (ours) |
|:--------------:|:------------:|:---------------:|:-----------:|:------------------------:|
|**FID**           |     23.014    |       20.495     |   21.373   |              18.450           |
| **Precision** |     0.657       |      0.837       |   0.765     |               0.886             |
| **Recall**      |     0.351       |      0.635       |   0.842     |               0.856             |
| **RMSE**      |     6.650      |       6.520       |   6.780    |               6.450              |

**Table 2: Performance on the LFW dataset.**  
|                      |     WGAN   |  Topo-GAN   |   DDPM    |  Topo-Diffusion (ours) |
|:--------------:|:------------:|:---------------:|:-----------:|:------------------------:|
|**FID**           |     19.820     |           -          |   13.155   |                7.873          |
|**Precision**  |     0.593       |           -          |   0.595     |               0.864           |
|**Recall**       |     0.574       |           -          |   0.733     |               0.973           |
|**RMSE**       |     6.875       |           -          |  4.6011    |               2.974           |

**Table 3: Performance on the NASA’s Satellite Imagery.**
|                      |     WGAN   |  Topo-GAN   |   DDPM    |  Topo-Diffusion (ours) |
|:--------------:|:------------:|:---------------:|:-----------:|:------------------------:|
|**FID**           |     31.020    |           -          |   10.796   |             6.362          |
|**Precision**     |     0.657       |          -           |   0.789     |             0.869             |
|**Recall**           |     0.351       |          -          |   0.875     |              0.891             |
|**RMSE**           |     37.893      |          -         |   13.959   |             7.909             |

We are still training TopoGAN model on LFW dataset, and NASA satellites imagery, and we will update them into the final version once completed.

Reviewer comment (Novelty):
The author(s) claim that “we introduce two persistence point transformation methods (see Eq. 2), i.e., (i) birth-lifespan transformation and (ii) differentiable distribution transformation”. These transformation methods “can transform a PD (i.e., a multi-set of persistence points) into a topological feature vector or function which can be easily integrated into any type of models”. However, all the transformation methods used in the paper are just standard Betti curve, persistent landscape, and persistent image. In fact, the so called “birth-lifespan transformation and differentiable distribution transformation” are just rename or summarization of existing methods. The vectorization of PD is a hot research area and various models have been proposed, including Barcode statistics, Algebraic functions and tropical functions, Binning approach, Persistent codebook, Persistent paths and signature features, 2D/3D representation, etc. Various topological kernels have also been developed. Detailed discussion of these models can be found in review papers such as:  
Pun, Chi Seng, Kelin Xia, and Si Xian Lee. "Persistent-Homology-based Machine Learning and its Applications--A Survey." arXiv preprint arXiv:1811.00252 (2018).  
Dey, T. K., & Wang, Y. (2022). Computational topology for data analysis. Cambridge University Press.  
Z. X. Cang, Lin Mu and Guo-Wei Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology, 14(1), e100592 (2018).  
The authors should not overclaim their contributions and should properly cite the related works.

Author response:
**Response to Question 1:** We thank the reviewer for suggesting these papers! We agree with this comment. Here we would like to point out that our proposed Topo-Diffusion is the first approach bridging persistent homology representations of the data and diffusion generative models (i.e., incorporates these two persistence point transformation methods into the diffusion model), and improves the robustness and performance on seven real-world and synthetic datasets. Also, we would like to highlight that our Topo-Diffusion provides flexible architecture which is capable of utilizing various types of topological summaries of the input object. We have cited all the above papers and introduced their methods/applications in the `Related Work’ section.

Reviewer comment (Theory):
The author(s) state that “A notable benefit of the birth-lifespan transformation is its parameter-free definition. This parameter-free Birth-lifespan transformation thus eliminates the necessity for any adjustments, negating the potential for overfitting”. This is not proper. The transformation can only generate a continuous function, which needs to be further discretized or vectorized. The size of the final topological vector depends on the process of discretization, and it can be very large if a refined mesh or grid is used. In contrast, if the Barcode statistics model or Algebraic functions are employed, the final topological vectors are always of limited sizes. In this way, they are really “parameter-free”.

Author response:
**Response to Question 2:** Regarding the "parameter-free", we mean that, compared with other deep learning models, it does not have arbitrary hyperparameters that the users have to set in order to perform the analysis. Now we have removed the "parameter-free" and modified this statement as "We introduce explicit constructions of topological summaries for persistence vectorizations through the birth-lifespan transformation using *Betti curve* and *persistence landscape*. Specifically, (i) Betti curve allows us to measure various topological information with a given filtration and does not inherit any implicit biases; and (ii) the advantages of persistence landscape are invertible, stable, and nonlinear." *(see Birth-lifespan transformation in the revision)*.

Reviewer comment (Presentation):
Figure 19 is very weird. The barcode is not consistent with the data. For instance, in the second simplicial complex (from left to right), it contains 7 0-dim homology generators, but the number of the corresponding 0-dim barcodes is just 6. Further, in the third simplicial complex (from left to right), it also contains 7 0-dim homology generators, but the number of the corresponding 0-dim barcodes is just 1. Moreover, it contains only 2 1-dim homology generators, but the number of the corresponding 1-dim barcodes is 3.

Author response:
**Response to Question 3:** Thank you for raising this question. To better illustrate the barcodes based on a sequence of Vietoris-Rips complexes for a point cloud, we have drawn a new figure on a point cloud with 5 points *(please refer to the Figure 19 in the revision)*. As shown in Figure 19, the top three subfigures are snapshots of the evolving complex as threshold $\epsilon$ increases. Note that the positions of the points in the figure do not exactly reflect the interpoint distances. The distances between two points in the point cloud less than or equal to $\epsilon$ are depicted in blue. The $\epsilon$ values corresponding to the two ends of the horizontal bars mark the birth and death times of topological features. We can observe that (i) when $\epsilon=0$, $\beta_0 = 0$ (i.e., $H_0$) and $\beta_1 = 0$ (i.e., $H_1$); (ii) when $\epsilon=0.4$, $\beta_0 = 3$ and $\beta_1 = 0$; and (iii) when $\epsilon=0.8$, $\beta_0 = 1$ and $\beta_1 = 3$.

Reviewer comment (Theory):
Page 4, “where $K$ denotes the maximum dimension of simplices of $x_t$”. Normally the dimension of PD can be different from the highest order of simplices.

Author response:
**Response to Question 4:** We agree with this comment. Following the Reviewer’s suggestion, we have fixed this sentence, i.e., where $K$ denotes the maximum dimension of homological features of $x_t$, and updated it in the revision.

Reviewer comment (Writing):
The paper is not very clearly written and could do with one more round after this one (if it does not make it this time).

Author response:
Thank you for the feedback. As suggested, we have revised the manuscript by including more details and refining the writing of the manuscript. Specifically, we have included more explanations and references of PH/TDA throughout the manuscript (including introduction, related work, preliminaries, and methodology sections). We also would like to highlight our new interpretation of the diffusion process in diffusion models from a perspective of structural information degrading and reconstruction and how those structural information can be represented by PH/TDA., included in the main body of the revised manuscript. In the introduction section, we have summarized the diffusion process from topological degradation and reconstruction. We explained the relationship between the forward process and topological feature degrading. We further apply multiple widely-used TDA features generation methods to analyze persistent homology information in diffusion. In the reverse process, we conduct the same TDA generation approach to reconstruct the image. We have discussed the technique in detail in the experiments section and demonstrated its effectiveness via a straightforward visualization on point clouds and images. We have added multiple quantitative results on various datasets to evaluate the performance of our Topo-Diffusion model.

Reviewer comment (Theory):
We need to include this information in the forward and back processes in diffusion.

Author response:
Thank you for the feedback. As suggested, we have revised the manuscript by including more details and refining the writing of the manuscript. Specifically, we have included more explanations and references of PH/TDA throughout the manuscript (including introduction, related work, preliminaries, and methodology sections). We also would like to highlight our new interpretation of the diffusion process in diffusion models from a perspective of structural information degrading and reconstruction and how those structural information can be represented by PH/TDA., included in the main body of the revised manuscript. In the introduction section, we have summarized the diffusion process from topological degradation and reconstruction. We explained the relationship between the forward process and topological feature degrading. We further apply multiple widely-used TDA features generation methods to analyze persistent homology information in diffusion. In the reverse process, we conduct the same TDA generation approach to reconstruct the image. We have discussed the technique in detail in the experiments section and demonstrated its effectiveness via a straightforward visualization on point clouds and images. We have added multiple quantitative results on various datasets to evaluate the performance of our Topo-Diffusion model.

Reviewer comment (Writing):
Forward process: list and qualitatively explain the technical challenges and the approach taken.

Author response:
Thank you for the feedback. As suggested, we have revised the manuscript by including more details and refining the writing of the manuscript. Specifically, we have included more explanations and references of PH/TDA throughout the manuscript (including introduction, related work, preliminaries, and methodology sections). We also would like to highlight our new interpretation of the diffusion process in diffusion models from a perspective of structural information degrading and reconstruction and how those structural information can be represented by PH/TDA., included in the main body of the revised manuscript. In the introduction section, we have summarized the diffusion process from topological degradation and reconstruction. We explained the relationship between the forward process and topological feature degrading. We further apply multiple widely-used TDA features generation methods to analyze persistent homology information in diffusion. In the reverse process, we conduct the same TDA generation approach to reconstruct the image. We have discussed the technique in detail in the experiments section and demonstrated its effectiveness via a straightforward visualization on point clouds and images. We have added multiple quantitative results on various datasets to evaluate the performance of our Topo-Diffusion model.

Reviewer comment (Experiments):
Reverse reconstruction: do the same.

Author response:
Thank you for the feedback. As suggested, we have revised the manuscript by including more details and refining the writing of the manuscript. Specifically, we have included more explanations and references of PH/TDA throughout the manuscript (including introduction, related work, preliminaries, and methodology sections). We also would like to highlight our new interpretation of the diffusion process in diffusion models from a perspective of structural information degrading and reconstruction and how those structural information can be represented by PH/TDA., included in the main body of the revised manuscript. In the introduction section, we have summarized the diffusion process from topological degradation and reconstruction. We explained the relationship between the forward process and topological feature degrading. We further apply multiple widely-used TDA features generation methods to analyze persistent homology information in diffusion. In the reverse process, we conduct the same TDA generation approach to reconstruct the image. We have discussed the technique in detail in the experiments section and demonstrated its effectiveness via a straightforward visualization on point clouds and images. We have added multiple quantitative results on various datasets to evaluate the performance of our Topo-Diffusion model.