HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell data

We thank the reviewer for their constructive feedback on HiPoNet. We appreciate your recognition of our empirical evidence. The concerns raised are primarily about theoretical results of HiPoNet. We address your concerns and questions as follows:

Regarding missing Appendices

As noted by the other reviewers, the appendices are indeed included in the supplementary material and contain theoretical exposition and discussions of limitations of our model.

Applying Absolute Value

One of the key features of our method is adoption of techniques from the Simplicial Scattering Transform $5$ . This Simplicial Scattering Transform, which is a generalization of the graph scattering transform (which is in turn a generalization of Stephan Mallat’s scattering transform on Euclidean data from 2012 $1$ ) utilizes a cascade of alternating linear operations in the form of multiscale wavelet transforms for message aggregation, and absolute value non-linearities. These have shown to be successful when carefully crafted in various geometric domains including graphs $2, 3$ , manifolds $4$ , and simplicial complexes $5$ . Here, we go beyond $5$ by constructing multiple different simplicial complexes, each of which provides a different view of the data. We note that recently in $10$ , it was shown that one can also construct scattering-based methods which utilize other, more common, activations such as ReLU. Upon revision, we will clarify this point, i.e., that the absolute values may be replaced by other activation functions, and add more background on scattering based methods in order to improve the readability of the paper for readers not familiar with the scattering transform.

Connection between Heat flow and Diffusion

The construction of our wavelets is motivated by $8$ , which introduced Diffusion maps, a well-known manifold learning algorithm based on the Markov-normalized Gaussian diffusion kernel, as well as the graph scattering transform literature $2, 3$ . There is indeed a deep connection between diffusion kernels and heat kernels $6,7$ . When applied to finite datasets sampled from a compact Riemannian manifold embedded in Euclidean space, the heat kernel can be approximated by a diffusion matrix, which serves as a first-order Taylor approximation of the heat flow $6,7$ . Subsequently, $9$ used this diffusion matrix to define Diffusion Wavelets which could be used to construct a multiresolution analysis of such data sets and $3$ showed that these wavelets could be incorporated into a geometric scattering transform. Like $3$ , we use the graph random walk matrix as a discrete analog of the heat-kernel due to the fact that the heat-kernel describes the transition probabilities of a Brownian motion (i.e., the continuum analog of random walks). Our theory is meant to illustrate the descriptive power of the heat kernel and therefore motivate methods based on the random walk operators, its discrete proxy. We also note that we could follow the lead of $4$ and use wavelets more directly related to the heat-kernel, of the form $\\exp(-2^{j-1}L\_k)- \\exp(-2^jL\_k)$, with the exponentiation defined in the spectral domain. However, this would increase both the computational cost and memory requirements of our methods. We will add discussion to clarify the connection between diffusion operators on graphs and heat kernels on graphs.

Use of Random Walks on Oriented Simplicial Complex

In Equation (1), for completeness, we have a general definition of a simplicial complex Laplacian that, depending on the incidence matrix, can be oriented or unoriented. We also introduce an associated diffusion or random walk operator in both cases. However, we wish to clarify that, in our experiments, we always use the unoriented Laplacian for all $k \> 0$, and correspondingly, the associated random walk matrices are also on the unoriented simplices. The only exception is for k \= 0, where we use the standard graph Laplacian, which is a special case of the oriented Hodge Laplacian with arbitrary orientations. Thus, Theorem 4.4 also uses the oriented Hodge Laplacian, but restricts to the case of $k=0$. We never use higher order oriented random walks with “directionality ignored” as the general random walk definition implies. We will clarify this in text to avoid confusion.

The connection between heat flow and manifold distances comes via Varadhan’s formula applied to the graph Laplacian which is the same as the (oriented) Hodge Laplacian $k=0$. This connection is established in works such as $6$ and $7$ .

Use of Integer Coefficients in the Non-Oriented Case and Implications for Topological Properties

For non-oriented simplices, the entries of $B\_k$ are merely integers (not $\\mathbb{Z}/2\\mathbb{Z}$) and so the boundary operator does not square to zero. Thus these operators cannot be directly used for persistence homology or other TDA computations but would rather have to be binarized to do typical TDA operations, as the reviewer notes. However, this is not needed for our neural networks. Instead, we just use these operators to keep track of simplices and define diffusions on them.

HiPoNet Captures Known Biology

When we highlight features that align with known biology, our aim is to demonstrate that HiPoNet can recover biologically meaningful signals, which serves as an important validation of the method. This does not imply a reverse logic; rather, it shows that at a minimum, the model is capturing known biology. Beyond that, the model also might assign high importance to other features not currently annotated as biomarkers. The relevance of these biomarkers can only be shown through experimentation. However, these features represent potential novel discoveries that warrant future investigation. Part of the utility of HiPoNet is in uncovering such features, which may provide new biological insights. We believe there is no logical flaw in the current interpretation, and we will revise the section to make this framing more explicit.

We appreciate your feedback and hope that our responses have addressed all your concerns. Please let us know if there are any remaining questions or areas that would benefit from further clarification, as we are happy to provide additional details. If you believe all your comments have been satisfactorily answered, we would be grateful if you consider updating your score. We thank the reviewer again for your time and constructive review!

$1$ Mallat, Stéphane. "Group invariant scattering." Communications on Pure and Applied Mathematics 65.10 (2012): 1331-1398.

$2$ Gama, Fernando, Alejandro Ribeiro, and Joan Bruna. "Diffusion scattering transforms on graphs." arXiv preprint arXiv:1806.08829 (2018).

$3$ Gao, Feng, Guy Wolf, and Matthew Hirn. "Geometric scattering for graph data analysis." International Conference on Machine Learning. PMLR, 2019.

$4$ Chew, Joyce, et al. "Geometric scattering on measure spaces." Applied and Computational Harmonic Analysis 70 (2024): 101635.

$5$ Madhu, Hiren, Sravanthi Gurugubelli, and Sundeep Prabhakar Chepuri. "Unsupervised parameter-free simplicial representation learning with scattering transforms." Forty-first International Conference on Machine Learning. 2024.

$6$ Huguet, Guillaume, et al. "A heat diffusion perspective on geodesic preserving dimensionality reduction." Advances in Neural Information Processing Systems 36 (2023): 6986-7016.

$7$ Jones, Iolo. "Diffusion geometry." arXiv preprint arXiv:2405.10858 (2024).

$8$ Coifman, Ronald R., and Stéphane Lafon. "Diffusion maps." Applied and computational harmonic analysis 21.1 (2006): 5-30.

$9$ Coifman, Ronald R., and Mauro Maggioni. "Diffusion wavelets." Applied and computational harmonic analysis 21.1 (2006): 53-94.

$10$ Chew, Joyce, et al. "Manifold Filter-Combine Networks." arXiv preprint arXiv:2307.04056 (2023).

We thank the reviewer for their constructive feedback on HiPoNet. We appreciate that you found the proposed multi-view and higher-order learning framework compelling. The concerns raised are primarily clarification-related. We address your concerns and questions as follows:

Computational Complexity of HiPoNet

We present the computational complexity of HiPoNet in Appendix Section I. We show that HiPoNet’s complexity increases polynomially with the number of simplices. The primary computational bottlenecks in HiPoNet arise from two sources: (1) memory usage and runtime overhead associated with constructing and storing multiple higher-order simplicial complexes, and (2) parallel processing of multiple views, each requiring a separate pass through the model. Memory usage grows with both the order of simplices (e.g., triangles, tetrahedrons) and the density of the Vietoris-Rips complex, which is controlled by the ε-threshold. To mitigate this, we limit the maximal simplex order (typically to 2-simplices), use sparse representations, and apply low ε values to induce sparsity. While incorporating higher-order simplices (e.g., triangles) increases expressivity (see Table A8), using K \= 1 — which reduces the model to a standard graph—still yields strong performance and offers a practical configuration for large-scale datasets. Although multiple views do add computational complexity, they provide consistent performance improvements (Table A4), making the trade-off worthwhile in many cases. Overall, while HiPoNet introduces more overhead than standard GNNs, it remains tractable for current biological cohorts consisting of high dimensional datasets and offers tunable configurations for more scalable deployment.

Sensitivity to Hyperparameters

To assess the sensitivity to the threshold, we conducted a sensitivity analysis on the Vietoris–Rip’s threshold (As shown in the Appendix Table A6), which governs the construction of higher-order simplices, on the single-cell classification datasets.

To assess the effect of kernel bandwidth, we performed a sensitivity analysis by varying the kernel bandwidth used during graph construction. As shown in the Appendix Table A7, performance on the Melanoma dataset improves steadily with increasing bandwidth and peaks at 1.00 (90.90 ± 4.92), after which it slightly declines. On the PDO dataset, however, the best performance is observed at 0.75 (65.20 ± 1.30), suggesting that overly large bandwidths may oversmooth the neighborhood structure in some domains.

In our ablation experiments on single-cell datasets (as shown in the Appendix Table A4), we find that the number of views is a crucial hyperparameter of HiPoNet and that increasing the number of views generally yields better learning performance. We suppose this is because multiple views allow the model to capture increasingly rich information about a point cloud’s topology. In our ablation, we found that four views were best for both the melanoma and PDO datasets. However, we note that increasing the number of views may result in diminishing returns after a point, presenting a trade-off between marginal learning improvement and increased computational cost.

Application to non-Biological data

While we can apply HiPoNet to non-biological and 3D point cloud datasets, HiPoNet is designed for high-dimensional point clouds, such as single cell proteomics and transcriptomics data. Each point in these datasets corresponds to a cell that carries a rich set of features, and the global structure of the point cloud is important to the task. The model’s strength lies in handling heterogeneous features that are found in biological data. In more standard 3D point clouds such as object recognition, reweighting or transforming the features could distort the spatial structure of the object, which is crucial for such tasks. Furthermore, while our method can be applied to 3-D point clouds, there are many methods that are specifically designed for that setting. The power of our method is rooted in its ability to handle higher dimensions and heterogeneous point clouds, making it well-suited for biological data.

Overall:

We appreciate your feedback and hope that our responses have addressed all your concerns. Given that these comments do not require major revisions we urge you to increase your score. Please let us know if there are any remaining questions or areas that would benefit from further clarification, we are happy to provide additional details. We thank the reviewer again for your time and constructive review!

We thank the reviewer for their constructive feedback on HiPoNet. We appreciate that you found the proposed multi-view learning framework compelling. The concerns raised are primarily clarification-related. We address your concerns and questions as follows:

Custom differentiable Vietoris-Rips implementation

We modified the existing torchph package (which is differentiable persistence computations with respect to persistence features) to be differentiable with respect to the simplicial structure. In our implementation, for each view $k$, we compute a weight matrix $\\mathbf{W}\_k$, which defines the edge weights in the Vietoris-Rips complex. These weights are treated as input-derived quantities with gradients. We construct a PyTorch Geometric (PyG) data object using $\\mathbf{W}\_k$, with edge (or higher-order simplex) weights. The key step is the Simplicial Wavelet Transform (SWT), implemented via PyG message passing over the simplicial complex. Since this process depends on $\\mathbf{W}\_k$, and all operations in SWT are differentiable, the final output $\\Phi$ is differentiable with respect to the input. This enables the full pipeline to be trained end-to-end via backpropagation.

Computational Complexity of Simplicial Complexes

We acknowledge that simplicial complex construction and message passing can be computationally intensive, especially for large graphs. To mitigate this, lower $\\epsilon$-thresholds can be used to sparsify the complexes and restrict computation to lower-order simplices (e.g., up to 2-simplices), which significantly reduces overhead. Additionally, our implementation leverages batching and sparse tensor operations in PyTorch Geometric to scale efficiently. In practice, we found the runtime to be manageable. Furthermore, as shown in Appendix Table A8, including higher-order structure (e.g., K \= 2) improves performance in the majority of cases. These gains highlight the utility of simplicial representations in capturing subtle dependencies that pair-wise relations in graphs alone may miss. Thus, while higher-order modeling adds some cost, it offers tangible performance benefits and remains practical for the scales of biological datasets we target.

Application to non-biological datasets

While HiPoNet can be applied to non-biological and 3D point cloud datasets, it is designed for high-dimensional point clouds, which are abundant in biology due to the profusion of single-cell proteomics and transcriptomics data. The model is built to handle heterogeneous features common in high-dimensional biological data, a setting that is increasingly common with emerging single-cell and spatial omics technologies. In standard 3D point cloud tasks like object recognition, reweighting or transforming the features could distort the spatial structure of the object, which is crucial for such tasks. Although HiPoNet can operate in such settings, there exist models tailored for 3D geometric tasks. The design choices in HiPoNet target high-dimensional and heterogeneous data, making single cell data an appropriate testbed.

Clustering and Logistic Regression Baselines

We thank the reviewer for the suggestion. However, standard machine learning techniques such as clustering and linear regression are not directly applicable in our setting, as we do not use fixed-length feature vectors for our “observations”. Our observations are entire datasets (and hence we coined the term “data cohort” to describe sets of datasets). Thus our observations, which are individual datasets, are high-dimensional point clouds consisting of different numbers of points. HiPoNet is specifically designed to learn meaningful and rich vector representations from these point clouds in a way that it captures their underlying geometric structure, such that a neural network can provide classification and regression to these clouds. For the types of tasks that we focus on, geometric baselines (which we compare against) are more suitable baselines.

Optimal number of views

Thank you for the question. In our ablation experiments on single-cell datasets (as shown in the Appendix Table A4), we find that the number of views is a crucial hyperparameter of HiPoNet and that increasing the number of views generally yields better learning performance, until all the types of cellular processes are encoded by each view. Increasing the number of views also enables the model to capture the complex topology of the underlying point cloud. In our ablation, we found that four views were best for both the melanoma and PDO datasets. However, we note that increasing the number of views may result in diminishing returns after a point, presenting a trade-off between marginal learning improvement and increased computational cost.

Hyperparameter selection

For choice of hyperparameters such as $\\epsilon$-threshold for Vietoris-Rips and kernel bandwidth, we follow a few practical rules of thumb when applying the method to new datasets. The number of views often corresponds to the number of distinct cellular processes or distinct feature spaces, though additional views can be added to capture different distance metrics or scales. The $\\epsilon$-threshold for Vietoris-Rips generally depends on the density of the points in the space, and is typically set to a value (e.g., 0.50) that captures local geometry while avoiding overly dense complexes. For low-dimensional data, the $\\epsilon$-threshold would be much lower. For the kernel bandwidth, a standard practice is setting it to the distance to the $k$-th nearest neighbor $1$ . These guidelines have worked reliably across datasets, though minor tuning may still be beneficial depending on dataset scale and noise.

Overall:

We appreciate your feedback and hope that our responses have addressed all your concerns. If you believe all your comments have been satisfactorily answered we would be grateful if you increased your score. Please let us know if there are any outstanding concerns. We thank the reviewer again for your time and constructive review!

$1$ Moon, K. R., Van Dijk, D., Wang, Z., Gigante, S., Burkhardt, D. B., Chen, W. S., ... & Krishnaswamy, S. (2019). Visualizing structure and transitions in high-dimensional biological data. Nature biotechnology, 37(12), 1482-1492.

I thank the authors for the rebuttal and I acknowledge their responses to my questions.

We thank the reviewer for their constructive feedback on HiPoNet. We appreciate that you found the proposed multi-view learning framework compelling. The concerns raised are primarily clarification-related. We address your concerns and questions as follows:

HiPoNet Mitigates Oversmoothing

We thank the reviewer for their point on oversmoothing. To empirically examine whether our modeling choice mitigates oversmoothing, we conducted a comparative analysis of Dirichlet energy across various methods. As established by $1$ , Dirichlet energy provides a principled measure of the expressiveness of node representations, where lower values indicate oversmoothed embeddings, and higher values reflect greater feature variation across graph neighborhoods.

We computed the Dirichlet energy of node representations produced by different models. The results (mean ± standard deviation) are:

These results indicate that our model, HiPoNet, exhibits substantially higher Dirichlet energy than traditional message-passing networks. This supports the claim that HiPoNet mitigates oversmoothing and retains richer, more expressive node-level information. Notably, the GCN and diffusion-based models exhibit particularly low energy, consistent with oversmoothed representations. Thus, this empirical evidence reinforces the motivation for using wavelet transforms, which allow HiPoNet to capture multi-scale variation and higher-frequency components in the signal that are typically suppressed in standard GNN architectures.

Dataset Clarification

In the melanoma dataset, our task is to predict response to therapy, not recurrence. The dataset contains 54 patients, with 23 responders and 31 non-responders, resulting in a class distribution of 31:23. We perform 5 stratified folds, ensuring balanced representation of responders and non-responders in each fold. For spatial transcriptomics datasets (e.g., breast, prostate), we adopt the patient-level splitting protocol proposed in $2$ . Each split ensures that the test set contains patients not seen during training, thereby avoiding information leakage across spatial spots or slides derived from the same individual. This is critical to evaluating generalization performance. To evaluate generalization performance and mitigate overfitting, we use stratified splits across all tasks and report average metrics across multiple runs. Additionally, we employ regularization (e.g., dropout, weight decay) in all models. The consistent performance across held-out samples and repeated splits supports the claim that the models generalize well and are not overfitting to training data. In the PDO dataset, the sampling unit is cultures where each patient can contribute multiple organoid cultures, resulting in 1,678 cultures. Each culture is treated in triplicate with either vehicle control or titrated combinations of clinical therapies as described in $3$ . We thank the reviewer for bringing this up and will update Appendix G to clarify these points.

Mis-aligned Appendix references

Thanks for pointing this out. We will fix this in the revised manuscript.

We appreciate your feedback and hope that our responses have addressed all your concerns. Please let us know if there are any remaining questions or areas that would benefit from further clarification, we are happy to provide additional details. If you believe all your comments have been satisfactorily answered we would be grateful if you consider updating your score. We thank the reviewer again for your time and constructive review!

$1$ Cai, C., & Wang, Y. (2020). A note on over-smoothing for graph neural networks. arXiv preprint arXiv:2006.13318

$2$ Wu, Z., Trevino, A. E., Wu, E., Swanson, K., Kim, H. J., D’Angio, H. B., ... & Zou, J. (2022). SPACE-GM: geometric deep learning of disease-associated microenvironments from multiplex spatial protein profiles. bioRxiv, 2022-05.

$3$ María Ramos Zapatero, Alexander Tong, James W. Opzoomer, Rhianna O’Sullivan, Ferran Cardoso Rodriguez, Jahangir Sufi, Petra Vlckova, Callum Nattress, Xiao Qin, Jeroen Claus, Daniel Hochhauser, Smita Krishnaswamy, and Christopher J. Tape. Trellis tree-based analysis reveals stromal regulation of patient-derived organoid drug responses. Cell, 186(25):5606–498 5619.e24, 2023. ISSN 0092-8674.