HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell data
Authors: Siddharth Viswanath
Hiren Madhu
Dhananjay Bhaskar
Jake Kovalic
David R Johnson
Christopher Tape
Ian Adelstein
Rex Ying
Michael Perlmutter
Smita Krishnaswamy
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Deep Learning
Point Clouds
Multi-View Learning
High-Dimensional Data Analysis
Single-Cell Data
Primary area: machine_learning_for_sciences

Abstract: In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Our work is motivated by single-cell data which can have very high-dimensionality --exceeding the capabilities of existing methods for point clouds which are mostly tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e., one data set for every patient), necessitating models that can process large, high-dimensional point-clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric and topological information. In contrast,  HiPoNet models the point-cloud as a set of higher-order simplicial complexes, with each particular complex being created using a reweighting of features. This method thus generates multiple constructs corresponding to different views of high-dimensional data, which in biology offers the possibility of disentangling distinct cellular processes. It then employs simplicial wavelet transforms to extract multiscale features, capturing both local and global topology from each view. We  show that geometric and topological information is preserved in this framework both theoretically and empirically.  We showcase the utility of HiPoNet on point-cloud level tasks, involving classification and regression of entire point-clouds in data cohorts. Experimentally, we find that HiPoNet outperforms other point-cloud and graph-based models on single-cell data. We also apply HiPoNet to spatial transcriptomics datasets using spatial coordinates as one of the views. Overall, HiPoNet offers a robust and scalable solution for high-dimensional data analysis.

HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell data
Siddharth Viswanath*
Hiren Madhu
Dhananjay Bhaskar
Jake Kovalic
David R. Johnson
Christopher Tape*
Ian Adelstein
Rex Ying!
Michael Perlmutter
Smita Krishnaswamy
Yale University
Boise State University 3 Uniiersityy College London
"Equal Contribution
Correspondence: smita. krishhaswamyyyall..e. edu
Abstract
this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on nigh-dimensional point clouds. Our work motivated by single-cell data which can have very high-dimensionality exceeding the capabilities of existing methods for point clouds which are mostly tailored for 3D data. Moreover, modern single-cell and spatial experiments for every patient), necessitating models that can process large, high-dimensional point-clouds scale. Most current approaches build single nearest-neighbor graph, discarding important geometric and topological information. In contrast, HiPoNet models the point-cloud as a of higher-order simplicial complexes, with each particular complex being created using reweighting of features. This method thus generates multiple constructs corresponding different views of high-ding nigh-dimensional data, which in biology offers the possibility of disentangling distinct cellular processes. then employs simplicial wavelet transforms to extract multiscale features, capturing both local and global topology from each view. We show that geometric and topological information is preserved in this framework both theoretically and empirically. We showcase the utility of HiPoNet on point- cloud level tasks, involving classification and regression of entire point-clouds data cohorts. Experimentally, and graph-based models on single-cell data. We also apply HiPoNet to spatial transcriptomics datasets using spatial coordinates HiPoNet offers a robust and scalable solution for high-dimensional data analysis.
yield entire cohorts of datasets (i.e., one data set
find that HiPoNet outperforms other point-cloud
of the views. Overall,
Introduction
High-dimensional point clouds, i.e., sets of points A' = 1 S I now arise in many fields- most prominently single-cell analysis [55 45 12 34 31I. where using modern technologies such as mass cytometry scRNA-seq, large cohorts of patients can now be measured producing several high-dimensioonl data. Further, technologies like perturb-seq enable the possibility of studying single-cell data under many conditions leading to comparable cohorts of datasets. Thus, machine learning techniques that once reasoned about data points and classified data points, now have to reason about collections of datasets. This serves the motivation for HiPoNet illustrated in Figure []]
Generally, methods like UMAP 36 or PHATE 38 reduce point clouds defined by single-cell data into individual graph learned from all features (often in the tens of thousands). However, this single
39th Conference on Ncural Information Processing Systems (NcurIPS 2025).

---

graph may not specifically organize cells according to processes defined by subsets of dimensions. For instance, cells may be in a specific phase of the cell cycle which would be revealed by an organization based on cell cycle genes, or at a certain point in a differentiation process which would be revealed by stem cell genes. To address these limitations, HiPoNet utilizes multiple reweighted feature views. Additionally, existing neural network methods for point cloud data, such as PointNet 4 and its
variants [44 63 588 33] are primarily designed for 3D point clouds and rely on spatially localized features" Hence, they are not well equipped to handle high- nigh-dimensional point clouds or disentangle processes from high-dimensional data. Therefore, there is a growing need to develop neural networks that can efficiently handle these diverse high-dimensional point clouds, ingest them seamlessly, encode the cellular processes, and perform various downstream machine learning tasks on them.
In HiPoNet, we use multiple views of the data, each of which is learned using a feature reweighting vector, thus potentially detangling and making implicit biological processes explicit. Furthermore, rather than modeling the data as graph, we model it as a simplicial complex that captures higher- order relationships between cells, which HiPoNet analyzes using multiscale wavelets. Overall, we gain rich representation that disentangles processes by capturing hierarchical relationships and separating overlapping biological processes, while preserving geometric and topological information of the underlying point clouds. Preserving this structure is important because it reflects the intrinsic geometry and topology of the data, providing a better interpretation of complex biological processes, and hence improving downstream analysis.
To capture the global structure of the point clouds, we use a wavelet-based multiscale mes- sage aggregation scheme rather than the mes- sage passing operations utilized in most com- mon graph and simplicial neural networks. This choice is based on the observation that such networks struggle to capture long-range depen- dencies and multiscale relationships as well as work on the geometric scattering transform
RO
Figure HiPoNes pipeline.
[5]] showing that wavelets may help remedy this issue. Although newer models like graph transformers [6] can capture long range dependencies, they often essentially ignore the graph geometry. When applied these point clouds, they only work single graph and are do not preserve the intrinsic geometric structure of the point cloud. However, using diffusion able to show theoretically that we capture the underlying geometry of the point cloud including the O-homology (connected components) and curvature. Empirically we show that we can predict persistence homology directly from our simplicial neural network, without explicitly computing topological signatures or reducing the data to those features. These properties that are preserved by HiPoNet are essential in its ability to perform classification tasks, such as response to immunotherapy or drug treatment response. The key features of HiPoNet include:
[9 20 29 55
& 62
wavelets,
Learning multiple views the the data: We introduce learnable scaling mechanism to reweight each feature via a weighting vector, which we then use to learn a graph to organize the data by the processes represented by the particular view.
Higher-order constructions: We learn simplicial complexes from the data views. Our neural networks utilize these complexes and retain geometric and topological information including volume, curvature, connected components, and holes in the data.
Multiscale Message Aggregation: Rather than simple message passing schemes that smooth data, we use multiscale simplicial wavelets to aggregate features over the simplicial complex and construct simplicial scattering coefficients. This allows us to extract both local and global information from the data without oversmoothing.
We demonstrate the utility of HiPoNet on three diverse single-cell data cohorts, each comprising ensembles of datasets collected from multiple patients or conditions. The first involves melanoma [42 patients undergoing immunotherapy, where the task is to predict treatment response from multiplex ion beam imaging (MIBI) with 61K cells from 54 patients. The second features vatient-derived organoids [45 (PDOs) from 12 colorectal cancer patients, grown under various treatments and co-culture conditions: comprising of 1.8M cells, and the task is to identify the applied treatment. The third cohort uses spatial transcriptomics data from around 500 cancer biopsies [6]] capturing 40
We discuss these techniques more in Appendix B

---

protein markers per cell through CODEX technology along with spatial arrangements, comprising of about 558K cells across three data cohorts. These datasets highlight the scalability of HiPoNet for complex, high-dimensional biological point clouds, where complexity arises from the number of datasets rather than the number of points per dataset
2 Background
In this section, present the mathematical foundations needed for method. We model point clouds = .x,,..., J Jo as simplicial complexes in order to capture higher-order geometric and topological structures. We will first introduce simplicial complexes and their algebraic representations, including boundary matrices and Laplacians. We then describe wavelet transforms and diffusion processes for extracting multi-scale topological features, and finally discuss random walks as efficient approximations of diffusion for scalable computation.
Given a finite set X' X2, X.............. J. }. k-simplex, denoted as o = (xog. X x is subset of X containing A: points. If mk- a subset of as that contains points, say that - is a face of or.- A simplicial complex S is a finite collection of simplices that satisfies the principle of inclusion 49]. That is, we have Tk S whenever if 7k-1 a face of σ for some oR S. The order of simplicial complex, K. is determined by the highest-order simplex contained within S. Therefore K -th order simplicial complex S contains simplices of orders k = ....., K. Observe that when K = 1, simplicial complex is essentially equivalent to a graph, where the O-simplices and 1-simplices correspond to the vertices and edges.
Features of simplices across all orders are denoted as ( X Xx), with X E N where N₂ is the number of le simplices and Dy is the feature dimension of k-simplices. We note that we can consider both unoriented simplices or oriented simplices, where the orientation is defined via fixed ordering on the vertices, i.e., O-complexes. (For details, please (48] and the references within.) Except when otherwise specified, our theory and algorithms will apply to either case.
In simplicial complex S, k-simplex 0 S can have four types of neighbors:
Boundary adjacent neighbors faces: These are the (A: sisipllies Tk-1 that faces of ok. denoted as ((s( = (Tk-1 : -h- C ).
Coboundary adjacent neighbors co-faces: These are all (k + s)-simplices Tk+1 which have Ty as a face, denoted as .c((aa) { : C Tk+1).
Lower adjacent neighbors: These are simplices of the same order 0x that share common face, Pk-1, represented as NN(( (Tk Pk- = dj O Tke 1pk-1|= k 1-
Upper adjacent neighbors: These are simplices, тh of the same order as a that are contained in a common (k 1i-simplex pK+1 E S, given by (Tx) {TK pk+1 a Tke |p*+| k + 1}.
We let (ox) =
(ox)) denote the 1-hop neighborhood of simplex da, and for m 2 2, we define the m-hop neighborhood of a simplex is recursively as (ox) = UreAsm. (( for m 2.
The relationships between k-simplices and their faces are encoded in the boundary matrices, denoted as B e X where N₂ the number of simplices in S of order k. The (i.j)-th entry of B& is non-zero if the i-th (k 1) -simmllex is face of the j-th k-simplex. For oriented simplices, these entries B, take values +1, reflecting their relative orientation. For unoriented simplices, all of the non-zero entries of B, are equal to I. We let B = [B [B&K jmm denote the set of all boundary matrices.
We note that the boundary matrices defined in terms of only boundary adjacent the and boundary adjacent neighbors. However, they may also be used to construct the Laplacians which encode information about lower and upper adjacent neighbors. Specifically, we define the lower and upper Laplcians by L B. Bg and L
B B We then define the k-Hodge Laplacian by:
A L L
(with L8 Li 0). We note that for oriented simplices, Ao is merely the standard graph Laplacian.
3

---

2.1 Heat Diffusion and random walks on Simplicial Complexes
The heat equation on k-simplices is given by:
dux( öt
(2
Analogous to classical notions of heat diffusion, it describes how the distribution of heat propagates over the simplicial complex over time. The use of A, ensures that the diffusion process respects the higher-order connectivity of the complex, encoding its topological structure.
The intuition behind much of our methods is use the heat equation to uncover the topological structure of our point clouds and simplicial complexes, analogous to manifold learning algorithms such Diffusion Maps [4]. However, in our actual algorithm. we will use random walk matrices, which we interpret as computationally efficient proxies for heat diffusion (similar to [14] which uses a diffusion matrix as a discrete approximation of the heat kernel on the underlying data manifold). Analogous to heat equation, random walks allow information to spread, i.e., diffuse, over the simplicial complex. However, they allow for efficient implementation via sparse matrix-vector multiplications.
We note that simplicial random walks extend the concept of traditional random walks from graphs to simplicial complexes. enabling a richer representation of the data which is able to encode higher-order features. Unlike standard graph-based random walks, which model transitions between vertices, simplicial random walks capture transitions between k-simplices, thereby incorporating higher-order interactions. Formally, the simplicial random walk is represented by the transition matrix P&. which describes the probability of transitioning between k-simplices. For unoriented simplices, we define:
P = A
where D; - diag( AA is a diagonal matrix that normalizes the transition probabilities, ensuring that the columns of Ps, sum to one. In the case of oriented simplices, defining Ps is non-trivial since the entries of A& may be either positive or negative. Here, for the sake of simplicity, we will define Pk the same way as in the unoriented case, but with first taking the entrywise absolute values of As. i.., Pk = A D2 = diag((Ax (| [1). Alternatively, in the case k = 0 or 1, one could define Pk in terms of a suitably normalized version of the Hodge Laplacian. (For details, please see (50].)
3 Methodology
We now describe the proposed method, HiPoNet, High-dimensional POint cloud neural NETwork designed to learn expressive representations of an input point set X' {x1, X,, xn E Rd. As summarized in Algorithm
[ our framework comprises three primary steps:
1. Multi-view creation via learnable feature weighting: We learn multiple sets of weights a (+), 1 v < V, to highlight different feature dimensions to create reweighted point clouds p0 which are effectively different views of the data.
2. Simplicial complex-based point-cloud modeling: From each {v)), we construct Vietoris- Rips complex ((r) [57]. This approach allows us to capture not only pairwise relationships but also higher-order interactions within the data.
3. Multi-scale Feature Extraction via Wavelets:
For each simplicial complex S(e), use wavelets to construct simplicial scattering coefficients which form a multi-scale embeddings ()()), that capture both local and global relationships. These embeddings are then concate- nated into single representation Tb and are passed multilayer perceptron to yield the final predictions y.
By combining learnable multi-view feature weighting, simplicial complex construction, and simplicial wavelet transforms, HiPoNet leverages multiple perspectives of the underlying point cloud, at multiple scales to deliver robust representations for downstream classification and prediction.
Multi-view creation via learnable feature weighting: In many scenarios, such as scRNA-seq, feature vectors can encode critical properties, such as gene expression levels. However, not all dimensions (genes) contribute equally to downstream tasks, and irrelevant or noisy features can

---

qp
AGGR(W?, up W
ppa
ML
Feature Learning
Simplicial Wavelets
dP
Views
View
View
View
Feature Importance
Figure 2: (A) The HiPoNet architecture. (B) Feature weight visualized across three learned views
Algorithm 1: HiPoNet
Input Point cloud X, kernel bandwidth a, number of views V. Learnable Parameters: learnable weights (x()), MLP z.
Output: Iutput:Predicttes Predictions y.
for u to do
(1)) = (*) @ x, for 1 < n:
Ln ((()) = {x] T
3
((r) = Vietoris-Rips Rips (*())) :))
Compute boundary matrices ((d) from simplices in S(v);
Set x(") (w) {(v)
// Vertex features;
for k 0 to
do
L x (v)
// Features for higher-order simplices
k+1
for k = 0 to K do
10
Compute transition matrix: for = to to do
=
(0)
D
11
( (P/² (0)
21
(p(o) 25-1
x
12
13
Compute Scattering Coefficients:
S = 1 (v.)
x(") and S
| (x (nnn) (w.) (()))(1))
14
(S(B(r) x())) 1100 (1)
x
15
Aggregate features across simplex orders:
16 Final Aggregation and Prediction:
17 A ((B())x(()))
y z
19 return y
impair model performance, especially in high-dimensional settings. To address this, we introduce learnable weighting mechanism that dynamically adjusts the weight of each feature dimension. Coneretely, define V distinct weight vectors ("),, For the v-th view, each point x, is rescaled to
V. each parameter of the model.
(v) (v)
x, = [a] (v) Jal, 02 (w) 3.2. ... Od rid);
(v)
where @ denotes the Hadamard (elementwise) product. let p {x] (x(") the reweighted point cloud. The reweighting operations amplify the more informative dimensions for the task at hand while downplaying less relevant features, effectively reducing the need for manual feature engineering. In the context of single-cell data, where the feature space can be vast,
X denote

---

such automated reweighting may be crucial for isolating key gene expressions that drive improved downstream performance. The learned weights can also be used for other biological insights, such as which gene markers are important for cancer diagnosis as illustrated in Figure 2
Simplicial learning from piont cloud data For each reweighted set P()), we create a simplicial complex S(Ø), To do this, we first define kernelized distances between two points x(") and xe)") as:
-||x (0)
exp
20²
We then fix scale parameter, e, and construct the Vietoris-Rips Complex 8(v), a simplicial complex defined by the rule that 0% {x x x!!) is an element of S(v) if
a!"" I for all 1 S p, q S k. After constructing the simpicial complex StD), we construct the boundary operators
B(a), We then define feature matrices by X!") p( (i.e., the i-th row of x(") is x x(") and then
x k-1 ) k. ) In our code, we use a custom implementation of the Vietoris-Rips filtration that enables us to make simplicial construction differentiable, making HiPoNet end-to-end trainable.
(0)
We repeat this process V times, and construct (² for each view 1 S S V. By constructing an ensemble of simplicial complexes ((1),8t2 ( our method integrates V distinct perspectives of the original high-dimensional point cloud. Constructing a simplicial complex S(v) for each reweighted set (1()) allows the model to disentangle and analyze distinct subspaces or biological processes captured by each view. For instance, each complex may focus on different cellular process, capturing unique local and global structures that is overlooked when relying on a single representation. Consequently, subsequent learning stages benefit from a richer of structural cues, enhancing the overall representation quality for downstream tasks.
S(V),
Multi-scale feature extraction via simplicial wavelets After the simplicial complexes have been constructed, we can leverage geometric deep learning methods to compute representations for the point cloud. Although message passing neural networks (MPNNs) for simplicial complexes 66 9 23 have been introduced, they often suffer from oversmoothing effects 4 where representations become indistinguishable after multiple layers. In addition, oversquashing sesssag-passsing frameworks. As the receptive field each node grows, large amounts of information from distant nodes are compressed "squashed" into a fixed-size vector, leading to loss of important information between the nodes. This limits the ability of MPNNs to model long-range dependencies.
is another constraint of such
In order to address these shortcomings, we employ the simplicial wavelet transform [35] 46 47)
(SWT). The SWT is a multi-scale feature extractor for processing a signal X defined on the simplices. These wavelets are based on diffusion wavelets introduced in [13] which utilize random walk matrices at different time scales. In SWT, the random walk matrices are calculated first as per Eq 3 Then, features are iteratively aggregated over all the neighborhoods and stored for each scale j. Finally, difference between different scales is taken to extract multiscale features.
The random walk diffusion matrix plays crucial role in the SWT. For each view we interpret each
(P()) x()) as diffused node features at scale j. Our wavelets, defined below, can thought of as measuring the differences between these diffused features at different scales. Alternativly, from signal processing perspective, these wavelets can be interpretted as a band-pass filters, with each j highlighting different frequency bands within the signal.
5 S J. is defined the difference between Formally, the simplicial wavelet transform at scale j. 5. diffusion at consecutive scales:
w (e5)
x (v)
After applying SWT for multiple scales j. we apply non-linearity, which we choose to be the absolute value |-], inspired by the geometric scattering transform [22 66
41], a multilayered feedforward network constructed using diffusion wavelets graphs. We then repeat this process with second wavelet, at a different scale j' > j. and then another absolute value. This yields first- and oecond-order simplicial scattering coefficients of the form SKbix( (o)
| (e))
(1) and S (2)

---

I [ (e.)) I. We then denote the collection of all first- and second-order scattering coef-
ficients for each view by (S(B()) X (1)) ) x JosiEJ,
(v)
(
Jo After
IgkSK
ISASR
extracting features for each view we aggregate the features over views as: (ggr((((BB)) S, x
S(B (
then processed by a multilayer perceptron z to generate our final predictions, ý = z(
4 Theoretical results
In this section, we provide theoretical motivation for our model. Specifically, we first show that heat diffusion on simplices captures the O-homology of the point cloud. Then, we show that simplicial complexes can be expanded into graph, and the heat equation on the simplicial complex agrees with the heat equation on the equivalent graph. Then, we show that the diffusion operators can capture geodesic distances. Proofs, as well as more detailed theorem statements, are provided in the appendix.
Heat Diffusion and Connectivity. A simplicial complex said be connected if any two simplices can be linked by a sequence of simplices that share common faces, More formally, if for any two simplices a and r. there exists m 2 such that Nn (o)). If simplicial complex not connected, it can be decomposed into union of disjoint connected components, each of which is maximal connected subcomplex.
The following theorem demonstrates that the solution to the heat equation on a simplicial complex remains confined the connected components where the initial condition is supported. This reflects the intuitive idea that heat cannot diffuse across disconnected regions of the complex. It also aligns with the concept of O-homology in algehraic topology since the connected components of simplicial complex correspond to the generators of its O-homology group. Thus, the heat equation's confinement connected components ensures that diffusion dynamics respect the 0-homology structure of the simplicial complex. The proof of Theorem is available in Appendix C
Theorem 4.1. The heat equations, Eq respect the O-homology structure of the simplicial complex.
Proof Sketch. The heat equation is governed by As, which acts locally by coupling neighbors. This ensures that heat diffusion cannot propagate between connected components.
Heat Diffusion on Simplicial Graphs. In this section, we demonstrate that the solution to the heat equation on simplicial complex S agrees with the solution the heat equation on an associated simplicial graph, i.e., graph where the vertices represent simplices of all orders and edges encode upper and lower adjacencies. This construction allows us to equate heat diffusion on the simplicial complex as heat diffusion on an associated graph, providing simplified framework for analysis. The proof of Theorem 4.3 is in Appendix D
Definition 4.2 (Simplicial Graph). Let S be simplicial complex. The associated simplicial graph G(S) is graph whose vertices are given by Vgg) = and whose edge set Eg)) consists of pairs of simplices which are either upper lower adjacent, defined in Section 2 that is.
F(g) {(a,o'} o, E V()), a' E NN(o) (N)(
Theorem 4.3. The heat equation on a simplicial complex agrees with the heat equation on the associated simplicial graph GS), defined in terms of Ag ((G(S)) where B(G(S) is the incidence matrix of (S)).
Proof Sketch. The graph Laplacian Ag on the simplicial graph can be represented as block diagonal matrix with the Hodge Laplacians A as the diagonal blocks.
Heat Solution Captures Geometry. Most of our experiments focus on single-cell data which are known satisfy the manifold hypotheses 228 24 17 55 D. That is, the data points x, lie upon some low-dimensional manifold. More specifically, we interpret each of the transformed point clouds (((0) as corresponding to different submanifolds of some global cellular manifold. The following theorem establishes that the diffusion operator on the simplicial complex can approximate geodesic distances these underlying manifolds. The proof of Theorem 4.4 is in Appendix E
7

---

Theorem 4.4. Assume that a transformed point cloud X lies upon a Riemannian manifold M. Then, the 0-th order Hodge Laplacian Sn on the associated oriented simplicial complex can approximate geodesic distances on the underlying manifold.
Proof Sketch. The result follows from Varadhan's 's formula [53] which shows that distances may be computed via the heat kernel, as well as Theorem 3 of [15] which analyzes the convergence of the graph heat kernel to the manifold heat kernel as well as Theorem 4.3 which relates the graph heat equation to the simplicial complex heat equation.
This theorem formalizes the link between diffusion processes, e.g., the heat equation, and the geometric structure of the data manifold. Notably, computation of geodesic distances on manifolds is computationally expensive and often infeasible in high dimensions. By using diffusion- based approximations, we can efficiently estimate geodesic distances with discrete operators. Additionally, we note that there is a long literature (dating back to at least (13) relating the Laplacian and the diffusion operator to the geometry of the data manifold, including quantities such as curvature 77). Most relevant to our work are the results which relate the asymptotic expansion of the trace of the heat kernel to geometric quantities like dimension, volume, and total scalar curvature (see for instance (37)) Indeed, following immediately from Theorems 4.3 and 4.4 we have:
The equivalence between the heat kernel on S and G(S) implies the equivalence of dimension, volume, and total scalar curvature on the respective underlying data manifolds.
Corollary 4.5.
The proof of Corollary 4.5 available in Appendix Section F. We use Weyl's law [60] and the eigenvalue comparison theorem to prove the equivalence.
5 Empirical Results
We compare the performance of HiPoNet with KNN-GNNs (i.e., GNNs on KNN graphs) as well as PointNet++ and it's variants are described in Appendix H.T The KNN-based graph neural networks
(GCN [30] SAGE (25]. GAT (54), GIN
663 and Graph Transformer [16). TopoGNN (26)) are
presented in the first block. followed by point cloud and topology oriented models (DGCNN (58). PointNet++ 44 Poin(Transformer [65), TopoAE 39 GCNN 40), HGCNN (8). and finally the
proposed HiPoNet. We present the mean and standard deviation over 5 seed
2 A bolded score
indicates the highest overall performance, whereas an underlined value identifies the second-best performance. Unless otherwise stated, we use unoriented simplices in our experiments. Details on the data cohorts considered in our experiments are provided in Appendix G Hyperparameters are described in Appendix H We provide a discussion of computational complexity in Appendix [ Discussion of limitations and societal impact provided in Appendix Sections [ [ respectively.
Topology and Geometry Prediction. We evaluate HiPoNet on the task of predicting persistence features of the point clouds, which provides a topological summary of the dataset. The ground truth for persistence features was calculated by calculating using persistence diagrams. We note that the purpose of these experiments not to develop new method of computing persistence features. Instead, it is to show that HiPoNet has the expressivity needed in order to compute such features, indicating that
viable method for tasks which require network to capture the underlying topology of the data.
The results in Table 1 indicate that HiPoNet outperforms baseline methods, achieving the lowest MSE across both datasets. Notably, KNN-based models perform competitively, with KNN-SAGE and KNN-GIN showing relatively lower errors compared to other GNN- ap- proaches. However, point-cloud blaud- models (DGCNN, PointNet++, and (oinfrransfrmmer) exhibit significantly higher error values, suggest- ing that they struggle to capture the necessary
Model
Mclanoma
KNN-GCN
L0683 = 0.002
1.0454 = 0.018
KNN-SAGE
0.734 0.031
1.0338 + 0.010
KNN-GAT
1.101 4: 0.028
1.068 0.008
KNN-GIN
0.061
0.006
KNN-Grrahh www.Grrhhhrrnsfrrre
1.274 + 0.038
1.3754 + 0.006
DGCNN
25.412 0.001
1353,0262
1.12
PointNett+
28.418 + 0.001
28.417 + 0.005
PoinTTransfommer
0.014
HiPoNer
0.633 1 0.043
0,4046 + 0.006
Table 1: MSE ea prediction of persistence features.
'The code is available at at :/ttt:::///tthbb httm///tisshaaaaaaaaaaalllalleee
8

---

Data cohort
Nuumber cells
Total Datasets
Task
Data Modality
Melanoma palient samples
61K
54
Response immunsherapy
MIBI
Patient-derived organoids (PDOss
1.8M
1625
Treatment Administered
Mass Cytometry
196K
196
Outzome chemetherapy
Charville
196K
196
Cancer recurrence
UPMC
308K
308
Outcome chematherapy
CODEX
308K
308
Cancer recurrence
DFCI
54K
54
Outeome chemotherapy
Table 2: Information about single-cell data
topological features for persistence feature prediction these data sets, perhaps because they were designed with 3D point clouds in mind, rather than high-dimensional point clouds.
Single-cell data classification. Next we assess the performance of HiPoNet on classifiying point clouds from single-cell data cohorts, described in Table 2 (Detailed descriptions of this data is provided in the Appendix(G) As mentioned in the introduction, we emphasize that each data cohort is a measurement of thousands cells on a large cohort of patients and the label prediction of outcome (of disease or treatment). As we can see in Table 4 HiPoNet outperforms all the graph-based, topological, and point cloud methods in Melanoma. On the PDO data, it is second to TopoGNN. However, HiPoNet is much more consistent than TopoGNN having standard deviation of 0.94% in comparison to TopoGNNs standard deviation of 16.15%.
Application to spatial transcriptomics data. In addition to the experiments on single-cell data, we further demonstrate an application of HiPoNet in analyzing spatial transcriptomics data. Unlike previous experiments where multiple views were constructed in the same manner, here we construct two views by different methods: one capturing spatial proximity of cells and the other encoding gene expression similarity. Table 3 compares the predictive performance of various models on spatial transcriptomics data drawn from four different cohorts: DFCI, Charville, UPMC, and Melanoma. DFCI and Melanoma include one task-outcome prediction while Charville and UPMC includes two tasks eashereither outcome prediction recurrence prediction. Notably, HiPoNet achieves the top results in most settings, outperforming KNN-based graph neural networks (GCN, SAGE, GAT, GIN), PointNet++, and the KNN-GrappTTransfrrmet
Data
DFCI
Charville
UPMC
Melanoma Response
Task
Outcome
Recurrence
Outcore
Recurrence
KNN KNN-GCN GCN
0.597 + 0.049
0.547 + 0.010
0.642 + 0,056
0.668 + 0.032
0.5 + 0.0
0.606 0.11
KNN-SAGE
0,82 + 0.040
0.618 0.021
0,581 0.016
0.631 + 0.013
0.5
0.572 0.05
KNNN-GAT
0.557 + 0.047
0.675 + 0.051
0,530 + 0.032
0.647 + 0.029
0,5++0.0
0.567 0.05
KNN GIN
0.700 0.048
0.045
0.663
0.015
0,514 0.02
0.06
KNN-Graph Transooope Iransformer
0.568 1: 0.051
0.578 :: 0.040
0,5 2 0.0
0.629 1 0.00
0.5 : 0.0
0.528 : 0.04
PointNet++
0.491 0.057
0.451
0.078
0.499
0.001
0.506
0.01
0.495 0.00
0.503 0.00
HiPoNet
0.916 - 0.03
0.681 2 0.012
0.681 1: 0.01
0.665 + 0.01
0.6044 = 0.0
0.732 : 0.01
Table 3:
ACC ROC on Spatial Transeriptomics Classifictioon
Intepretability of Learned Features. HiPoNet offers high degree of interpretability enabled by its multi-view and learnable feature weighting architecture that learns meaningful representa- tions from high-dimensional point cloud data. As seen in the three bar plots Figure B)) each plot corresponds to different learned view or representation of the 30 protein expression markers in the melanoma single-cell classifica- tion task, with the y-axis including the learning importance of each marker in that specific view. Biologically meaningful markers like CD1lb, CDI18, and FOXP3 have consistently high importance across the different views, aligning with their roles in immune regulation within the tumor microenvironment and promoting immune evasion in tumors.
Model
Melanoma
PDO
KNN-GCN
72.72 + 5.45
0.70
KNN-SAGE
76.36 = 6.03
55.89 0.83
KNN-GAT
61.81 4.45
52,56 + 10.06
KNN-GIN
85.45 3.63
$7.02 1.20
KNN Gruph Framfforrre
8.90
TopoGiNN
88.18 8.19
79.90 =: 16.15
IIGNN
80000 7.6
DGCNN
63.33 12.47
40.00 + 4.36
PoiniNett+
22.42
PointTramsfommr
79.99 1 6.24
30.00 4.70
TopoAB
52.73
15.71
0.87
GCNN
63.63 + 0.0
29.34 1.03
HiPoNet
90.90 4.92
77.38 0.94
Table 4: Accuracies on classification tasks.
Mitigation of Oversmoothing. To empirically examine whether our modeling choice mitigates oversmoothing, we conducted a comparative analysis of Dirichlet energy across various methods. As established by [10), 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.
As seen in Table 5 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 standard GNN architectures.
Model
Dirichlet Energy
GCN
1.669 0.13
GAT
2.694 0.49
Graph Transformer
3.811
Diffusion- basedd GNN
0.584 0.00
0.00
HiPoNet (ours)
21,033 + 6.25
Table Dirichler Energy across different models.
components the signal that are typically suppressed in
Ablations. We provide ablations in Appendix ! Table A4 shows that increasing the number of views improves performance, with four views yielding the best results. In Table A5 we observe that removing multi-view learning or simplicial modeling leads the largest performance drops, confirming their critical importance. Table A6 presents sensitivity analysis of the Vietoris-Rips threshold, showing that the optimal threshold varies by dataset and significantly affects accuracy. In Table A7 we analyze the effect of kernel bandwidth, finding that careful tuning is required to avoid oversmoothing. Finally, Table A demonstrates that while 1st order simplices (essentially graphs) are sufficient for some data sets, incorporating higher-order simplices improves performance for others, particularly in predicting clinical outcomes.
Conclusion
In this work, we introduced HiPoNet, a novel neural network architecture designed for high- dimensional point cloud datasets. By leveraging multiple views, higher-order constructs, and multi- scale wavelet transforms, HiPoNet effectively captures complex geometric and topological structures inherent in biological data. Our experiments demonstrate that HiPoNet learns distinct, meaningful representations tailored to each dataset and significantly improves downstream analysis across diverse single-cell and spatial transcriptomics cohorts. These results highlight the importance of integrating higher-order and multi-scale information to overcome the limitations of existing point cloud and graph-based models.
7 Acknowledgements
D.B. received funding from the Yale Boehringer Ingelheim Biomedical Data Science Fellowship and the Kavli Institute for Neuroscience Postdoctoral Fellowship. M.P. acknowledges funding from The National Science Foundation under grant number 01A-2242769. S.K. is funded in part by the NIH CMMMSROIGM139999
ROI R01GM130847), NSF CAREER award IIS (1S-2047856, NSF IIS-2403317, NSF DMS--327211 and NSF CISE-2403317. S.K is also funded by the Sloan Fellowship FG-2021- 15883, the Novo Nordisk grant GRI 12933. S.K. and M.P. also acknowledge funding from NSF DMS-2327211.
References
[1] Constantin Ahlmann-Eltze and Wolfgang Huber. Analysis of multi-condition single-cell data with latent embedding multivariate regression. Nature Genetics, Jan 2025. ISSN 1546-1718, doi: 10.1038/<41588-022 001-5-88--2---096---1- URL https://:/// http::///ooooo////// org/ 10. 1038/ 020-888-------99------
[2] Uri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In International Conference on Learning Representations, 2021. URL https
/ooenreview / het/fforruiii--11000000
[3] Michael Angelo, Sean C Bendall, Rachel Finck, Matthew B Hale, Chuck Hitzman, Alexander D Borowsky, Richard M Levenson, John Lowe, Scot D Liu, Shuchun Zhao, al. Multiplexed ion beam imaging of human breast tumors. Nature medicine, 20(4):436-444 2014.
10

---

[4] Muhammet Balcilar, Guillaume Renton, Pierre Héroux, Benoit Gaüzère, Sébastien Adam, and Paul Honeine. Analyzing the expressive power of graph neural networks in spectral perspective. In International Conference on Learning Representations, 2021. URL https
[5] Sergio Barbarossa and Stefania Sardellitti. Topological signal processing over simplicial complexes. IEEE Transactions on Signal Processing, 68:2992-3007, 2020. ISSN 1941-0476. doi: 10.1 (099tsp.22....9819999 1920. URL htRlhhtp://dxx doi. org/ 10. 1109/TSP. 200000 2981920
[6] Claudio Battiloro, Lucia Testa, Lorenzo Giusti, Stefania Sardellitti, Paolo Di Lorenzo, and Sergio Barbarossa. Generalized simplicial attention neural networks, 2024. URL https
7/arxiv. org/ abs/2309 wwt.v/.rggabb//399.// 02138
[7] Dhananjay Bhaskar, Kincaid MacDonald, Oluwadamilola Fasina, Dawson Thomas, Bastian Rieck, lan Adelstein, and Smita Krishnaswamy. Diffusion curvature for estimating local curvature in high dimensional data, 2022. URL https://arxii org/ htt///a/.//2222/3// 03977
[8] Bernhard G Bodmann and fris Emilsdóttir. A scattering transform for graphs based on heat semigroups, with an application for the detection of anomalics in positive time series with underlying periodicities. Sampling Theory, Signal Processing, and Data Analysis, 22(2))17, 2024.
[9] Cristian Bodnar, Fabrizio Frasca, Yu Guang Wang, Nina Otter, Guido Montúfar, Pietro Liò, and Michael M. Bronstein. Weisfeiler and lchman go topological: Message passing simplicial networks. CoRR, abs/2103 2021. URL http:///aa.xo.oooo/bb////a/a//// 220///ab///13... 2103 03212
arxiv.
[10] Chen Cai and Yusu Wang. A note over-smoothing for graph neural networks, 2020. URL https://arxiv.ort org/ abs 13318
[11] Shiu-Yuen Cheng. Eigenvalue comparison theorems and its geometric applications. Mathema- tische Zeitschrift, 1975. URL httpp://doi org/10../000 1007 (007/BF01214381
[12] Joyce Chew, Holly Steach, Siddharth Viswanath, Hau-Tieng Wu, Matthew Hirn, Deanna Needell, Matthew D. Vesely, Smita Krishnaswamy, and Michael Perlmutter. The manifold scattering transform for high-dimensional point cloud data. In Proceedingy of Topological, Algebraic, and Geometric Learning Workshops 2022, volume 196 of Proceedings of Machine Learning Research, pages 67-78. PMLR, 25 Feb-22 Jul 2022. URL htttt:://proceedinig mlr. wwwpssvveeeeee v1ts//v16/////// satee222ahhml
[13] R. Coifman and M. Maggioni. Diffusion wavelets. Appl. Comp. Harm. Anal., 21(1):53-92 2006.
[14] Ronald Coifman and Stéphane Lafon. Diffusion maps. Applied and computational harmonic analysis, 21(1):5-30, 2006.
[15] David B Dunson, Hau-Tien Tieng Wu, and Nan Wu. Spectral convergence of graph laplacian and heat kernel reconstruction in 1 from random samples. Applied and Computational Harmonic Analysis, 55:282-332 22263,6,2222 2021.
[16] Vijay Prakash Dwivedi and Xavier Bresson. A generalization of transformer networks to graphs. CoRR, abs/2012.09699, 2020. URL htttttt////riiiiii wttp:///.w.rraa/////2... 09699
[17] Charles Fefferman, Sanjoy Mitter, and Hariharan Narayanan. Testing the manifold hypothesis. Journal of the American Mathematical Society, 2944):983-1049, 2016.
[18] Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks, 2019.
09401
[19] Fernando Gama, Alejandro Ribeiro, and Joan Bruna. Diffusion scattering transforms on graphs. arXiv preprint arXiv: 1806. 08829, 2018.
[20] Fernando Gama, Alejandro Ribeiro, and Joan Bruna, Stability of graph scattering transforms. Advances in Neural Information Processing Systems, 32, 2019.

---

[21] Feng Gao, Guy Wolf, and Matthew Hir. Geometric scattering for graph data analysis. In International Conference on Machine Learning. pages 2122-2131. PMLR, 2019.
[22] Feng Gao. Guy Wolf, and Matthew Him. Geometric scattering for graph data analysis. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference Machine Learning. volume 97 of Proceedings of Machine Learning Research, pages 2122-2131. PMLR, 09-15 Jun 2019. URL htttt:::/ppoceeeeeee
press srees/v97/ / mlr www......................mmmm..e
gao19e html
[23] L.
Giusti, C. Battiloro, P. Di Lorenzo, S. Sardellitti, and S. Barbarossa. Simplicial attention neural networks, 2022. URL httpss:///axx org/ aaa//203.00773533
[24] A. N. Gorban and I. Y. Tyukin. Blessing of dimensionality: mathematical foundations of the statistical physics of data. Philosophical Transactions of the Royal Society A: Mathematical,
Physical and Engineering Sciences, 376(1118)))1111112733333 2018. doi: 10. 2019////...21...00.1000
URL http::/:/popplllccceyyyllliiiiia wwwp:hhhininnnnggooooooooooo
1098/ rsta. 2037
[25] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning large graphs In Advances in Neural Information Processing Systems (NeurlPS), pages 1024-1034, 2017. URL U https: //tpx//aaxvvvv..a wwwpeiiioorraaaeeeee httpg//bb//770.... 02216
[26] Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau. Bastian Rieck, and Karsten Borgwardt. Topological graph neural networks, 2022. URL https::/aaxxv..rr//bbbb 07835
[27] Bobo Hua and Xin Luo. sawiies-gaffney-grieee yan lemma on simplicial complexes, 2017. URL /ttps://arxiv.. www.axiw.orggbb///.a//
[28] Guillaume Huguet, Alexander Tong, Edward De Brouwer, Yanlei Zhang, Guy Wolf, lan Adelstein, and Smita Krishnaswamy. A heat diffusion perspective geodesic preserving dimensionality reduction. In seirth- Conference on Neural Information Processing Systems, 2023. URL httpp////ooprreeeiieettttftrrrr/
[29] Yuanhong Jiang, Dongmian Zou, Xiaoqun Zhang, and Yu Guang Wang. Limiting over- smoothing and over-squa of graph message passing by deep scattering transforms. arXiv preprint arXiv:2407.06988, 2024.
[30] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference Learning Representations (ICLR), 2017. URL https: ftttps///arxiv/ org/ abs/1609 02907
[31] Charlotte Kröger, Sophie Müller, Jacqueline Leidner, Theresa Kröber, Stefanie Warnat- Herresthal, Jannis Bastian Spintge, Timo Zajac, Anna Neubauer, Alcksej Frolov, Caterina Carraro, Silka Dawn Freiesleben, Slawek Altenstein, Boris Rauchmann, Ingo Kilimann. Marie Coenjaerts, Annika Spottke, Oliver Peters, Josef Priller, Robert Perneczky, Stefan Teipel, Emrah Düzel, Frank Jessen, Simone Puccio, Anna C. Aschenbrenner, Joachim Schultze, Tal Pecht, Mare D. Beyer, Lorenzo Bonaguro, and DELCODE Study Group. Un- veiling the power of digh-dimensional cytometry data with cycondor. Nature Communica- tions, 10702, Dec 2024. ISSN 2041-1723, doi: /tttp:://do..org/0...
URL
[32] Ilya Loshchilov and Frank Hutter. Fixing weight decay regularization in adam. CoRR, abs/1711.05101,, 2017. URL http://rr/// owg.xivv..rg//ee abs http///a/1111111111
[33] Xu Ma, Can Qin, Haoxuan You, Haoxi Ran, and Yun Fu. Rethinking network design and local geometry in point cloud: A simple residual mlp framework, 2022. URL/httss://axxi org/
abs/ abs/2202. 2ttp/2222.000 07123
[34] Kincaid MacDonald, Dhananjay Bhaskar, Guy Thampakkul, Nhi Nguyen, Joia Zhang, Michael Perlmutter, Ian Adelstein, and Smita Krishnaswamy. A flow artist for high-dimensional cellular data, 2023. URLhttps: //tp/////vvv///////a///a////aav/ www./rgg///baaeeeee// abs//308.0000000
12

---

[35] Hiren Madhu, Sravanthi Gurugubelli, and Sundeep Prabhakar Chepuri. Unsupervised parameter- free simplicial representation learning with scattering transforms. Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, editors, Proceedings of the s International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 34145-34160. PMLR, 21-27 Jul 2024.
URL htttps://proceeedingg http:/eddiiia.......rre/////////
mlr. wwwpppresssvvee paress/v235/maeeee http//////mmaaa2aaaaaa/ html
[36] Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction, 2020. URL httts:://axxii org/ larg/bbe aas/ /abs//880..o 03426
[37] Ion Mihai. Chapter complex differential geometry. In Franki J.E. Dillen and Leopold C.A.
Verstraelen, editors, Handbook of Differential Geometry, volume 2 of Handbook of Dif-
ferential Geometry, pages 383-435. North-Holland, 2006. doi: https://doi.org wwt.ooor///..1111
$1874-5741106/800099-9, URL https://wwrt www.w..cinneeirec...om com/ www.///cecc///ae//cccccooccc///a article/ P11/$1874574106800999
[38] Kevin R Moon, David Van Dijk, Zheng Wang, Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew Hirn, Ronald Coifman, al. Visual- izing structure and transitions in high-dimensional biological data. Nature biotechnology. 37 (12):1422-192, 2019.
[39] Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt. Topological autoencoders. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 7045- 7054. PMLR, 13-18 Jul 2020. URL httpss:/pproceednggs mlr wttp./prrrss///11mo0000000 html
[40] Mathilde Papillon, Guillermo Bernárdez, Claudio Battiloro, and Nina Miolane. Topotune framework for generalized combinatorial complex neural networks, 2025. URL https
//axxiv.or/abbe /bg// 0
[41] Michael Perlmutter, Alexander Tong. Feng Gao, Guy Wolf, and Matthew Hirn. Understanding graph neural networks with generalized geometric scattering transforms. SIAM Journal on Mathematics of Data Science, 5(4):873-898, 2023.
[42] Jason Ptacek, Matthew Vesely, David Rimm, Monirath Hav, Murat Aksoy, Ailey Crow, and Jessica Finn. 52 characterization of the tumor microenvironment in melanoma using multiplexed ion beam imaging (mibi). Journal for ImmunoTTherapy of Cancer, (Suppl A): 2021. doi: 10.1 101116//--22211STTTCC 0252.. URL /ttpsttss:///ttc. bmj. com/ /omt/ett//ett//ppl 2/A59
[43] Charles R. Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation, 2017. URL httpt:///axxii wttpiivv..rr//b//////////// 00593
[44] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J. Guibas. Pointnet++: Deep hierarchical feature learning point sets in metric space. CoRR, abs/1706.02411 2017. URL / //arxiv.org www./ogg//bbbbaaeeeee wtb.///6.... 02413
[45] 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- wew. 11.005. URL
5619.e24, 2023. ISSN 0092-8674. doi:
h/wpp/:///www../. www.ecceieeetttcoommmm
[46] Naoki Saito, Stefan C Schonsheck, and Eugene Shvarts. Multiscale hodge scattering networks for data analysis. arXiv preprint arXiv:2311.1022 2023.
[47] Naoki Saito, Stefan C Schonsheck, and Eugene Shvarts. Multiscale transforms for signals on simplicial complexes. Sampling Theory, Signal Processing, and Data Analysis, 22(1):2, 2024.
13

---

[48] Anton Savostianov, Francesco Tudisco, and Nicola Guglielmi. Cholesky- -ike preconditioner for hodge laplacians via heavy collapsible subcomplex. SIAM Journal on Matrix Analysis and Applications, 45(4):1827-1849, 2024.
[49] Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaic. Random walks on simplicial complexes and the normalized hodge laplacian. CoRR, abs/ 807. 607.05044, 2018. URL uttttpt:///axii org/ abs/abo// 05044
[50] Michael T Schaub, Austin R Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaic. Random walks on simplicial complexes and the normalized hodge l-laplacian. SIAM Review, 62(2): 353-391, 2020.
[51] Alexander Tong, Frederik Wenkel, Dhananjay Bhaskar, Kincaid Macdonald, Jackson Grady, Michael Perlmutter, Smita Krishnaswamy, and Guy Wolf. Learnable filters for geometric scattering modules. arXiv preprint arXiv:2208.08 07455, 2022.
[52] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(86):257--2605, 2008. http://mm owg.oog/////// arg/papers/v9/ wandermaatenO8a. html
[53] S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Communications Pure and Applied Mathematicc, 20(2):431-455, 1967. doi:
URL
www.r..wwil................ com/
doi 01600000002
[54] Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. In International Conference on Learning Representations (ICLR), 2018. htRt::tts:///art org/ abs 1710
http//bb/111..... 10903
[55] Aarthi Venkat, Dhananjay Bhaskar, and Smita Krishnaswamy. Multiscale geometric and topological analyses for characterizing and predicting immune responses from single cell data. Trends in 4 44(7):551-56) 2023. ISSN 1471-4906. doi: https://doi..rr//0..
i0166)..t 003. https:/s:://wwt www.httpss//wws.cceeeddieee sciencedirect. cow/ woo//ccience/rrtiieo www.eennceaaricceppp....
$14714906220008444 Special issue: Systems immunology II.
[56] Matthew Vesely and David Johnson. Processed multiplexed ion beam imaging (mibi) of tumor microenvironments of 54 melanoma samples following immunotherapy. 2025. URL
/ttp::://aata.mendeleeyy mwndeley...eeooeoommommmoooo wwt.:cccccaaaaaaaa cctm///aaaaaetss797777777///
[57] Leopold Victoris. Über den höheren zusammenhang kompakter räume und eine klasse von zusammenhangstreuen abbildungen. Mathematische Annalen, 1927. httpss//:///a org/
10.1007/BF01447877
[58] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph cnn for learning on point clouds, 2019. /tt/sttss://axx org/ Abs/1801.07829
[59] Frederik Wenkel, Yimeng Min, Matthew Hirn, Michael Perlmutter, and Guy Wolf. Overcoming oversmoothness in graph convolutional networks via hybrid scattering networks. arXiv preprint arXiv:2201.08932, 2022.
[60] Hermann Weyl. Nachrichten von der gesellschaft der wissenschaften zu göttingen. Seahhematisch-Phyiikkliee
Klasse, 1911.
[61] Zhenqin Wu, Alexandro Trevino, Eric Wu, Kyle Swanson, Honesty Kim, H. Blaize D' A'Anio, Ryan Preska, Gregory W. Charville, Piero D. Dalerba, Ann Marie Egloff, Ravindra Uppaluri, Umamaheswar Duvvuri, Aaron T. Mayer, and James Zou. Space-gm: geometric deep learning of disease-associated microenvironments from multiplex spatial protein pro- files. bioRxiv, 2022, doi: 10. 10.1101/2002.05522249977 Content/earrly/022/ /00pall///222////2 2022. 05. 12. 491707
URL httph::////// h www. biorxiv.org!
14

---

[62] Charles Xu, Laney Goldman, Valentina Guo, Benjamin Hollander-Bode Bodie, Maedee Trank- Greene, lan Adelstein, Edward De Brouwer, Rex Ying, Smita Krishnaswamy, and Michael Perlmutter. BLIS-Net: Classifying and analyzing signals graphs. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pages 4537-4544 PMLR, 02-04 May 2024. URL https
/ //proceedings. mlr www..presss///ee http///222xxx2xcccc/ html
[63] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference Learning Representations (ICLR), 2019. URL
https://arxiv org/ abs/ /tbs//180... 00826
[64] Yao Yu Yeo, Precious Cramer, Addison Deisher, Yunhao Bai, Bokai Zhu, Wan-Jin Yeo, Mar- garet A Shipp, Scott J Rodig, and Sizun Jiang. A hitchhiker's guide to high-dimensional tissue imaging with multiplexed ion beam imaging. Methods in cell biology, 186:213, 2024.
[65] Hengshuang Zhao. Li Jiang, Jiaya Jia, Philip Torr, and Vladlen Koltun. Point transformer, 2021. httt:htps:://// //axtt wrw.
09164
[66] Dongmian Zou and Gilad Lerman. Graph convolutional neural networks via scattering. Applied and Computational Harmonic Analysis,
2020.
15

---

NeurIPS Paper Checklist
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
The abstract and introduction clearly state the key learning, simplicial modeling, and wavelet-based aggregation- which are directly supported by the theoretical results and experiments.
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: A dedicated Limitations section is provided in Appendix Section J It discusses strong assumptions such as the manifold hypothesis (Appendix Section E). sensitivity to hyperparameters like the kernel bandwidth and VR threshold (Tables A6 and A7 and the computational complexity of the method (Appendix Section T..
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed not penalize honesty concerning limitations.
3. Theory assumptions and proofs
16

---

Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: All theoretical results, including Theorem 4.4 and Theorem 4.3 accom- panied by explicit assumptions and full proofs in the Appendix (Sections) and D Proof
sketches are also included in the main paper to provide intuition.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: All experimental settings, including hyperparameters, model configurations, dataset details, and evaluation metrics, are fully described in Section H and Appendix [] Datasets used are publicly available (c.g., MIBI [42). PDO [45], and Spatial Transcrip-
tomies (61). Code is provided at hhttss://gt thub. com wtt.uu..co//krsssaasaaarrroo to facilitate full reproducibility.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution be necessary to either make it possible for others to replicate the model with the same dataset, or provide access the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
specific model and empirical evaluation, may
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm.
(b) If the contribution is primarily the architecture clearly and fully.
model architecture, the paper should describe
(c) the contribution a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
17

---

(d)
We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, described in supplemental material?
Answer: [Yes]
Justification: Code is available at httts:///githu: con//Krishmaawwwyyaaa/iiii
The repository includes detailed instructions for setting up the environment, running ex- periments, and reproducing all main results. All datasets used publicly available and referenced in Section G
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.cm cc/
cublic/guidest Seegl///uuddes/ooeuuuiiiiss
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results the new proposed method and baselines. If only subset of experiments should state which ones are omitted from the script and why.
reproducible, they
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: All training and test details, including data splits, hyperparameters, optimizer settings, and how they were selected, are provided in Appendix H The optimizer used is AdamW with a constant learning rate of 10 and weight decay of 10 All datasets and experimental configurations are clearly described, and additional reproduction details are provided in the released code repository.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can be provided either with the code, appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
18

---

Answer: [Yes]
Justification: All experiments report mean values along with standard deviation over independent runs with different random seeds. These results are reported all relevant tables (e-g., T Table 1 and clearly indicate the variability due to random initialization and data splits. We specify that the reported error bars are standard deviations, as noted in the experimental setup in Appendix
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by enror bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that out of range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: The experimental setup (Appendix HD specifies that all experiments were performed on single NVIDIA A100 GPU with 40GB memory. Each experiment took approximately 2-3 hours training time. Total compute usage for all experiments, includ- ing ablations and hyperparameter tuning, is estimated at approximately 400 GPU hours. Preliminary experiments that did not make into the final results required an additional 100 GPU hours.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics http::///euuipps.ccototo http:///wwrrg...ccooa/caccccaaaa
Answer: [Yes]
19

---

The research adheres to the NeurIPS Code of Ethics. All datasets used are publicly available and anonymized, ensuring no privacy violations. The proposed methods aim to advance scientific understanding without harmful societal impacts. Potential limitations and risks have been transparently discussed in Appendix L.
Justification:
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: The paper discusses the potential for positive societal impact in advancing our understanding of tissue microenvironments, which could aid biomedical research and precision medicine. Possible negative impacts, such risks of overinterpretation of spatial patterns without proper validation and privacy concerns with patient data, are acknowledged in Appendix K along with suggested mitigation strategies.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly. harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility ML).
11. Safeguards
Question:
Does the paper describe safeguards that have been put in place for responsible release of data models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Answer: [NA]
Justification:
The paper does not introduce models or datasets with high risk for misuse.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
20

---

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: All datasets used in this work are properly cited in the main text and Appendix.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g.. website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of in the package should be provided. For popular datasets, papprssiithheeeoooco.pap thcode. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: This paper does not introduce any new datasets or other assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: This paper does involve any crowdsourcing experiments or research involving human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
21

---

Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethies, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: This work does not involve any new data collection from human subjects; datasets used were publicly available, and necessary IRB approvals were obtained by the original data providers.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: LLM is used only for writing, editing, or formatting purposes.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy for what should or should be described.
22

---

A
Notations
Table A1:
Notations used throughout the paper.
Notation
Description
X {x1,
Xn
Original point cloud with data points
Input feature dimension (c.g., number of genes or markers)
V
Number of distinct views
(+1)
Rdd
Learnable weight vector for view v
((1)
Reweighted feature vector for point in view e Reweighted point cloud for view Gaussian Kernel distance between points i and in view v Scale parameter for Vietoris-Rip complex construction Simplicial complex for view
(())
(e)
S(e)
Maximum simplex order the complex
Ok
A s-iimplex in the complex
Number of k-simplices
(
E Nex
Feature matrix for k-simplices in view e
B (o)
Boundary matrix for order-k simplices view Hodge Laplacian for k-simplices in view Simplicial random walk matrix for k-simplices in view v Number of diffusion scales used in the scattering transform Simplicial wavelet scale for order-k simplices in view First-order scattering coefficients
A k
(
(ug)
(*)
Second-order scattering coefficients
S(B X
Collection of all scattering features for view
Aggregated feature representation across all views Multilayer perceptron used for classification
y
Final model prediction
B
Related Works
The vast majority of work point Moud-based learning has focused three dimensional problems, such as mapping and interpreting sensor readings in computer vision or localization tasks 43
Early methods such PointNet [43] and its successor PointNet++ [44] introduced permutation-invarian models with the ability to extract local and global features from 3D point clouds. Newer methods such Dynamic Graph Convolutional Neural Network (DGCNN) (58 leverage dynamic graph representations to better extract rich features from input point clouds, while PointMLP 333 and Point Transformer [65] use standard Multilayer Perceptrons and local self- attention mechanism as the building blocks for better performance in classification and regression tasks. We are primarily interested in biological domains which are much higher dimensional, ranging from dozens in proteomics datasets to thousands in transcriptomic datasets, we require new approach that not limited by architectural decisions and computationally efficient when working with figh--imenssonal data. Despite the advancements in these existing methods, they are fundamentally designed for point clouds and struggle with performance and scalability in a high-dimensional setting they rely on spatial heuristics, because local neighborhoods become less meaningful in high-dimension. As opposed this, HiPoNet able to scale to high-dimensional data while also preserving the geometry of the dataset.
65
33 58].
A1

---

Single-cell analysis has greatly benefited from graph-based methods that learns global structure from dige-iimenional data. These methods typically rely on a single graph construct to infer relationships between cellular states. Methods such as UMAP [6 and t-SNE 52 perform non- linear dimensionality reduction by constructing a neighborhood graph and embedding cells into low-dimensional space. On the other hand, PHATE (38 is a dimensionality reduction method that captures both global and local nonlinear structure but only constructs a single graph from the data. While these methods have been useful in understanding various biological processes, they may fail to organize cells based on processes governed by subsets dimensions with just a single connectivity structure. In contrast this, HiPoNet has the ability to model distinct cellular processes by leveraging its mueti-view framework, preserving important geometric properties.
C Proof of Theorem
The following Theorem is more detailed version of Theorem 4.1 which states that the heat equation respects with the O-homology of the underlying point cloud.
Theorem C.1 (Connected Components on Simplicial Complexes). Let S a simplicial complex of order K with m connected components, and let Σn be the set of k-simplices (k K). We partition Σg = Ex.1 LE J... J...LLkk.m. where each Lg.s corresponds to the s-simplices in a single connected component. Let S be subset k-simplices, and assume that S is the union of several connected components: S = Sk.i. Ll... ER..,- Assume the initial condition, ug(-, 0), of the diffusion equation
has support contained in S. Then, for any k--th order simplex 0 & S and for all t O, we have:
[ke t) = 0.
(5)
Proof. The Hodge-Lapllacan A acts locally, meaning i, j-th entry will be zero unless the i-th complex is a neighbor of the j-th k-complex (which can happen either via i = or via upper/lower adjacent neighbors). As a result, if the initial condition u(-.0) 0) has support contained in S, the solution u(ass must remain confined to S for all > follows:
0. To formalize this, define function ū(crk. t) as
(o t) = { uu (uk. t),
if E S, if S.
10
We will show that ( satisfies the same heat equation u(ok,t). First, consider simplex σ E S. By definition, (ok,)) u( for all 0, and since u (Tx,t) satisfies the heat equation, it follows that:
aū(ak, 1) Ou(ak, !) AAUUA(OR = - - 0t
Øt
where in the final equality we used the fact that A acts locally.
Next, consider a simplex de S. By definition, un (k (ū.((r..) O. Furthermore, since A& acts locally and 0% is not in S. we have also: A 8) = dt
for all 2 0, which implies
(since neighbors must be in the same connected component). Thus, 0ü (orl)) Axūu(ox,t) 0.
and so (o.(o,,t) satisfies the heat equation.
Lastly, since (ak, f) satisfies the heat equation for all simplices S and coincides with the initial condition u(-, 0) on S, follows from the uniqueness of solutions to the heat equation that in (Ox, t) un u f) for all é >> 0. Therefore, for any simplex a & S and for all t > O, we have: up (a 1) = ūn (ax, 1) = 0.
D Proof of Theorem
Below, we will state Theorem D.I which is more detailed statement of Theorem 4.3 First, we will introduce some notation and preliminaries.
We will assume that the vertices of G(S), (i.e., the simplices E S) are ordered in manner consistent with the size of the simplices, that is, all of the & simplices come before all of the + simplices in ordering. Let N₂ denote the number of simplicial complexes of order & and let
A2

---

N Ex=
k== N& denote the cardinality of S, i... the total number of simplices. For each t 2 0, let
Ng (, (, ER R denote the solution to the graph heat equation:
Oug (o.t)
Agug
(6)
O1
where Ag = B((()) is the graph Laplacian of G(S), and B(G(S) is the incidence matrix of (()). (If S oriented simplex, then (() is the signed incidence matrix, otherwise is the unsigned incidence matrix.) Let ug denote the solution to the heat equation associated to A k in Eqn. and, for t define the solution the simplicial complex heat equation, us(;t) E IR by
us(;, t) [uo t), ................................ ux ((x())))
(7)
(i.e., the denote the concatenation of all of the wx--)
Theorem D.1 (Agreement of Heat Equation Solutions). Let S be simplicial complex, and let ((S) be is associated simplicial graph as defined in Desimition 4.2) Let us(o, I) denote the solution the heat equation on S. (See Eqn 7 and also Equ. Similarly, let ug(a, denote the solution to the heat equation on G(S).
Assume that the initial conditions us(. (,, 0) and ug(:, 0) are consistent, i.e.
us(a, 0) ug(o, 0) for all V(G). Then, for all and for all simplices o E S, the solutions agree, i.e., we have us us( t) = ug (o, t).
Proof. Since the edge set of (SS) defined in terms of upper and lower adjacent neighbors, Ag can be written in block diagonal form as
A
0
0
A
0
A2
Ag
(8)
0
A
0
A
K
This allows us to see that us also satisfies the graph equation equation since
Ao
0
0
0
0
-Agus(,8)
Ms(: f)
-K-1
0
AAK
Ao
0
(uo(;;))
A
0
ui (:tt)
A2
0
(2(:)t)
:
0
-K--
0
0
uuk (tt)
A AK
Bugl
Ao uo(:,t)
A un (.)t)
AZ (2()
aust (..
0t
2
us(::).
Oe
Ax uk (:)
06
Therefore, ug us by uniqueness of solutions.
A3

---

E
The proof of Theorem 4.4
The following is more detailed statement of Theorem 4.4 which states that the diffusion simplices captures the geodesic distances.
Theorem E.1 (Capturing Geodesic Distance). Assume that a transformed point cloud X lies upon a Riemannian manifold M. Let S be an oriented simplicial complex constructed from from X(v), and let (SS) be its associated simplicial graph. Denole the 0-th order heal kernel on S by Hj(o, T) where T,T represent simplices of any order in S. So that Hjuo(-, 0) solves the heat equation with initial condition uo(,,0, Then, we may approximate the geodesic distance dy 7) on the manifold M from H.-
Proof. Let h2 denote the heat kernel on the underlying manifold. Varadhan's formula 4t log he (a, r) d(
[33
shows that
lim t-
Theorem 3 of [15] shows that the graph heat kernel H2 uniformly converges to the manifold heat kernel le and so the result follows from Theorem D.1 which establishes the equivalence of the graph heat equation and the heat equation on simplicial complexes.
F Proof of Corollary 4.5
The equivalence between the heat kernel on S and G(S) implies the equivalence of dimension, volume, and total scalar curvature on the respective underlying data manifolds.
Corollary 5.
Proof. Let hy denote the heat kernel on the manifold M. For any two points x, E M. The eigenvalues Xj and corresponding eigenfunctions Φ; of the Laplace-Beltrami operator determine the manifold heat kernel via the spectral expansion:
h4 v) = Σe e Act (.(I)((()))
1=0
The eigenvalues are influenced by the volume and curvature of the manifold via Weyl's law [60 and the eigenvalue comparison theorems [T]. Thus, as in the proof of Theorem 4.4 the result follows from Theorem 3 of [15].
G Data Cohorts
Melanoma: We use data originally collected by Ptacek al. 42 and made available in Vesely and Johnson (56), which consists of 54 melanoma patients, who underwent checkpoint blockade immunotherapy. The data contains approximately 489 to 1784 cells from the tumor mnicro-enviromeennt of each patient, resulting in a total of 1 11,862 cells. Each cell is characterized by 29 protein markers measured by Multiplexed Ion Beam Imaging (MIBI) 64 3 This data can be modeled as 54 point clouds in 29-dimensional space, with sample sizes ranging from 489 1784 points per point cloud. The learning task binary classification of whether the patient experienced recurrence not (including stable discase as a 'non-reeuuee noo-rreuure,ce", ), predicted from the protein expression levels. We note that MIBI is able to measure the spatial locations of the cells well as their protein counts. However, in our experiments, we do not use the spatial locations of MIBI. We make this choice that, with this data set, the high- nigh-dimensional point clouds corresponding to protein counts.
assess the ability of methods to learn purely from
Patient-derived Organoids (PDOs): We next consider data from Ramos Zapatero al. featuring patient-derived organoids (PDOs) from 12 different patients with colorectal cancer. Patient-derived organoids (PDOs) are cell cultures grown from patient tumor samples. Each culture, representing a PDO sample, is characterized by 44 proteins and contains approximately 1137 cells. Collectively, we consider 1,678 such cultures, each of which viewed as a 44-dimensional point cloud, resulting in total of 2 million cells. The goal is to infer treatment administered to the PDO based on cellular states, serving as synthetic task for our model.
A4

---

Data
Task
K
Classification
Melanoma
Persistence prediction
Classification
PDO
Persistence prediction
DFCI
Outcome
Outcome
Charville
Recurrence
Outcome
2
UPMC
Recurrence
Table A2: Order of simplicial complex
Model
Accuracy
GCN
41.67 + 25.00
GIN
38.89 7.86
Graph Transformer
50.00 + 16.67
HiPoNet
63.60 + 0.00
Table A3: Comparison of model accuracy on Spatial Data.
Spatial Transcriptomics (ST): We next consider Spatial Transcriptomics data generated using 40-plex CODEX (CO-Detection by Indexing) immunofluorescent imaging workflow from Wu et al. [61. capturing 40 protein markers per cell across datasets from UPMC, DFCI, and Charville, comprising 500 patients, each of whom either had head-and-neck cancer or colorectal cancer. Here we construct views in two different manners:
Spatial View: Cellular coordinates describing how cells are arranged in two-dimensional tissue sections.
Gene View: Dozens of markers or transcripts measured per cell, detailing complex biological states.
The overarching goal is to predict outcome of chemotherapy on cancer patients. Another goal is to predict cancer relaspe of recovered patients. This task particularly leverages HiPoNet's 's ability to fuse spatial context with transcriptional signals for clinically relevant insights in cancer research.
H Experimental Setup
The parameters of HiPoNet optimized using the AdamW optimizer [32 with constant learning rate of 10-1 and a weight decay of 10-1 We train the model for 100 epochs on every dataset for each model. We record the best metric (accuracy in classification tasks; mean squared error in regression tasks) achieved on the test set in each training run. We compare model performances on each dataset and task by the mean and standard deviation of their resulting five metric scores. For the spatial transcriptomics data, we use the folds from Wu et al. [61. In Table A2 we describe order of the simplicial complex that we have used for each data set. Typically, our setup necessitates approximately two to three hours of training time for each dataset on single NVIDIA A100 GPU,
H.1 Baselines
We compare the performance of HiPoNet on the classification tasks outlined in Section 5 to two groups of alternative methods. First, to demonstrate the expressive power of our multiview graph embeddings over traditional graph neural network techniques, we construct K-nearest neighbor graphs of the input point cloud data and attempt to learn a classifier using several popular graph neural network architectures. These include Graph Convolutional Networks (GCN) (3 GrE
25
Graph Attention (GAT) Networks [54] Graph Isomorphism Networks (GIN) 63 and Graph
Transformer Networks (GTNs) [16]. We discuss their performance on our dign-dimensional tasks in
Section 5
A5

---

Num. of Views
Melanoma
PDO
27.27 = 12.85
59.80
1.99
2
54.54 = 11.13
61.10 + 0.17
90.90 + 4.92
77.38 0.94
Table A4: Accuracies (mean + standard deviation) from ablation the number of graphs. Best result is bolded and second best is underlined.
Model
Melanoma
PDO
HiPoNet
90.90 + 4.92
77.38 + 0.94
w/o multi-view
27.27 + 12.85
59.80 1.99
w/o structure
46.08 + 8.38
14.09 + 1.49
w/o reweighing
56.36 + 7.60
48.30 3.99
w/o wavelets
70.90 + 14.93
41.53 1.84
Table A5: Ablation over components.
Second, we compare our method directly state-of-the art, bespoke point cloud learning methods. While DGCNN (58], PointNet++ [44], and Pannfrransorrmrr [65] all perform very well on the three-dimensional problems they were designed to tackle, as shown in Table 4 they struggle to successfully classify high-dimensional point clouds.
I Computational Complexity
The computational complexity of one step of diffusion process, performed via sparse multiplication, has the complexity of O Nx), where Dy is a dimension of the features of k-simplices. To construct the wavelets, we need to perform steps of diffusion for V views. Therefore, the cost of the SWT is O(V D2 Nz)). For the second order scattering coefficients there are (JJ) different choices of j and j'. Thus the cost is ( D&N&)) The complexity of reweighing through Hadamard product is O(D, No). The complexity of VR-filtration is O(N3). So, the total computational complexity of HiPoNet O(N3 V V
J Ablation
In Table A4 we present an ablation study of HiPoNet showing the impact of varying number of views. The results indicate that four views most effective for capturing the topology of the point clouds.
To assess the contribution of each component-multi and wavelet transform-we conducted an ablation study on the Melanoma and PDO datasets. As shown in Table A5 removing multi-view learning causes the largest performance drop (63.6% on Melanoma, and 17.6% on PDO), highlighting its central role. Additionally, we see that not using the structure of the data, but instead passing the point cloud features directly into an MLP classifier (without forming simplicial complex), leads severe degradation in performance on both tasks, with 63% drop on PDO and 44.8% drop on Melanoma. Finally, we see that removing the wavelet transform (i.e., using diffusion only) lowers performance across both datasets (with 20% drop on Melanoma and a 25.85% drop on PDO). These results confirm that each module plays a crucial role, with multi-view learning and structural modeling being especially vital.
learning, simplicial complex modeling,
To assess the sensitivity the threshold, conducted sensitivity analysis on the Vietoris-Rips threshold, which governs the construction of higher-order simplices, using the single-cell classification datasets. As shown in Table A6 performance varies with the threshold. On Melanoma, the best result is achieved at 0.50, while on PDO, a lower threshold of 0.15 yields the highest accuracy, indicating that the optimal choice can be dataset-specific. To select the appropriate threshold, we tune it until reaching critical beyot-beyond which the number of edges grows rapidly. These critical points are used our analysis. Notably, we found that PDO a critical point at 15, which leads to improved performance compared to what reported in the main text. These results suggest that
A6

---

Threshold for V-Rips
Melanoma
PDO
0.15
68.18 9.42
77.38 = 0.94
0.25
3.45::211
60.44 = 1.61
0,50
90.90 4.92
63.10 0.00
0.75
77.27 = 15.21
18.92=099
1.00
61.81 + 13.48
64.96 = 0.85
1.25
61.81 + 7.60
62.00 = 0.51
Table A6: Sensitivity analysis with respect to the Victoris- threshold.
Kernel Bandwidth
Melanoma
PDO
0.25
61.81 = 7.60
60.44 + 1.21
0.50
61.81 = 11.85
63.61 + 1.21
0.75
65.45 = 7.60
77.38 0.94
1.00
90.90 4.92
62.44 + 1.46
1.25
79.09 + 16.51
62.00 0.51
Table A7: Sensitivity analysis of kernel bandwidth (%).
the model is sensitive to this hyperparameter, and careful selection of the threshold important for achieving strong performance.
To assess the effect of kernel bandwidth, performed sensitivity analysis by varying the bandwidth used during graph construction. As shown in TAb performance on the Melanoma dataset initially improves with increasing bandwidth and peaks 1.00 (90.90 + 4.92), after which it slightly declines. On the PDO dataset, however, the best performance is observed at 0.75 (77.38 0.94), suggesting that overly large bandwidths may oversmooth the neighborhood structure in some domains. These results indicate that kernel bandwidth is a sensitive hyperparameter and should be tuned based on dataset characteristics. We will incorporate this analysis and its implications into the revised manuscript.
Table A8 presents results across four tasks tasks-PDO, PDO, Charville (Outcome and Recurrence tasks), and UPMC (Recurrence task) wish) K 1, 2, representing the maximum dimension of simplices used during complex construction. We observe that for PDO, performance peaks at = 1, suggesting that lower-order interactions are sufficient. In contrast, for outcome prediction on Charville and UPMC, higher-order structures (K 2) improve performance, particularly for the UPMC dataset where AUC increases from 0.538 0.6044. Notably, performance saturates or slightly drops K indicating diminishing returns from overly complex topology.
K Broader Impacts
This work introduces HiPoNet, neural network for high-dimensional point cloud analysis with applications in biological data such single-cell and spatial transcriptomics. Positive societal impacts include advancing our understanding of complex biological systems, enabling discoveries in discase mechanisms, and contributing to the development of more effective treatments through improved modeling gene and cell interactions. Potential negative impacts include the risk of over-reliance on model predictions in critical biological or clinical decision-making without sufficient experimental validation. Additionally, applying HiPoNet to sensitive biomedical datasets raises privacy concerns if appropriate safeguards are not maintained. We recommend that HiPoNet be used primarily for hypothesis generation rather than direct clinical application, with findings validated through rigorous experimental studies. Responsible data handling practices and collaboration with domain experts essential to mitigate these risks.
L Limitations
While HiPoNet advances the modeling of high-dimensional point clouds, has several limitations. First, although HiPoNet is relatively efficient, training very large datasets still requires substantial compute resources, especially for larger values of K. Second, the current model has been primarily validated biological datasets; its generalization other high-dimensional domains remains to be
A7

---

Dataset / Task (Metric)
==1
K 2
K =
PDO (Accuracy)
77.38 - 0.94
72.30 0.18
70.76 0.71
Charville (Outcome AUC-ROC)
0.55 0.001
0.598 + 0.12
0.539 - 0.05
Charville (Recurrence AUC-ROC)
0.681
0.681 0.01
0.681 0.01
UPMC (Recurrence AUC-ROC)
0.538 + 0.03
0.6044 = 0.0
0.583 = 0.01
Table A8: Performance across datasets and tasks for varying K.
explored. Finally, HiPoNet focuses on predictions rather than inferring causal relationships, limiting its direct applicability for causal inference tasks.
A8


Reviews:
[{"review_id": "wbvkMtVwb0", "reviewer": "Reviewer_kJJr", "summary": "This work presents HiPoNet, a method to model point cloud data using higher-order simplicial complexes. The method can represent the data using multiple views, learn interactions between these views through simplicial complexes, and aggregate features using multiscale wavelets. The approach is evaluated across a broad set of tasks, including classification of spatial transcriptomics data, demonstrating strong performance.\n\nWhile I found the biological motivation and empirical results compelling, I must acknowledge that I do not fully grasp the mathematical foundations behind simplicial complexes and wavelet transforms. My evaluation therefore focuses on the modeling motivation, empirical performance, and clarity of presentation.", "questions": "* Please double-check the Appendix references, e.g. L318: \"Detailed descriptions of this data is provided in Appendix F\". This should point to Appendix G instead. Similar for other appendices (e.g. Appendix I does not point to the limitations)", "limitations": "The limitations are presented in Appendix L", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strenghts**\n\n* The intuition of using multiple views of the data is compelling, e.g., scRNA-seq provides a high resolution picture of the active biological processes in each single-cell, and so having multiple views may allow the model to decouple multiple relevant processes for the task at hand. This is in contrast to existing dimensionality reduction algorithms such as tSNE and UMAP, which operate on a single view of the entire transcriptome in a task-independent way and so some features may therefore be convoluted and partially irrelevant for the given task.\n* The benchmark is fairly comprehensive and demonstrated results are impressive, i.e. HiPoNet seems to outperform existing methods across a broad range of tasks. In particular, I was surprised by the large margin demonstrated in the spatial transcriptomics tasks, which have direct clinical relevance. For example, HiPoNet seems to outperform the next best performing baseline by ~20% AUC ROC on the Melanoma response task. Nonetheless, the benchmark could be more transparent in terms of task description, data splits, and class imbalances (see Weaknesses).\n* The ablation results are compelling and help illustrate the contribution of key components, e.g. using multiple views.\n\n\n\n**Weaknesses**\n\n* In terms of the evaluation of the method, the data splits are unclear and, for some of the tasks (e.g. inferring patient recurrence) the total number of patients is fairly low (e.g. 54 melanoma patients). How was the data split and what are the class distributions? For the PDO task, i.e. inferring treatment administered to patient-derived organoids, the total number of patients is 12, while the total number of cultures is 1,678. What are the sampling units in this task (patients or cultures?) Was each culture treated in a different way? What are the treatments that the model is tasked to distinguish? Overall, could the authors show that the evaluation scores reflect improved generalization performance rather than overfitting? In summary, it is difficult to fully interpret the reported results without more transparency around dataset splits, class distributions, and evaluation units. A clear table summarizing each task (dataset, input modality, sample size, number of classes, class imbalance, and split strategy) would greatly improve clarity and interpretability.\n* This is not a weakness per se, but a request out of curiosity. The authors motivate the use of simplicial wavelet transforms arguing that more standard approaches like message-passing neural networks suffer from oversmoothing and oversquashing effects. Could this modelling choice be empirically demonstrated?", "was_revised": true, "final_justification": "I am satisfied with paper and the authors' response. Thank you for the additional oversmoothing benchmark and dataset clarifications.", "rebuttal": "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:\n\n### **HiPoNet Mitigates Oversmoothing**\n\nWe 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.\n\nWe computed the Dirichlet energy of node representations produced by different models. The results (mean ± standard deviation) are:\n\n| Model | Dirichlet Energy |\n| ----- | ----- |\n| GCN | 1.669 ± 0.13 |\n| GAT | 2.694 ± 0.49 |\n| Graph Transformer | 3.811 ± 0.80 |\n| Diffusion-based GNN | 0.584 ± 0.00 |\n| GWT | 15.807 ± 0.00 |\n| HiPoNet (ours) | **21.033 ± 6.25** |\n\nThese 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.\n\n### **Dataset Clarification**\n\nIn 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. \n\n**Mis-aligned Appendix references**\n\nThanks for pointing this out. We will fix this in the revised manuscript.\n\nWe 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\\!\n\n\\[1\\] Cai, C., & Wang, Y. (2020). A note on over-smoothing for graph neural networks. arXiv preprint arXiv:2006.13318\n\n\\[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.\n\n\\[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."}, {"review_id": "3vecP8gAHG", "reviewer": "Reviewer_WcXs", "summary": "Authors propose HiPoNet, an end to end differentiable neural net for regression, classification, and representation learning on higher dimensional point clouds. The key to their approach lies in modeling the data as a simplicial complex instead of a graph to capture higher-order relationships between cells. The authors showcase its applicability by modeling three different single-cell datasets (biological point clouds). Quantitative results showcase strong performance of the proposed approach.", "questions": "1. Have the authors tried a simple clustering-based approach as a baseline? What about a simple linear regression baseline?\n2. How did you decide the optimal number of views for feature weighting?\n3. From the ablation studies it seems that the methods requires careful tuning of number of views, and epsilon threshold. How do you suggest these parameters will change when a completely new dataset is used? What about kernel bandwidth?\n4. Can you elaborate on your custom differentiable Vietoris-Rips implementation? I'm trying to understand your end-to-end differentiable claim", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "strengths:\n- combination of using multi-view reweighting, simplicial complex and wavelets is novel and opens up applications for other areas like surface reconstruction\n- comprehensive baseline comparisons and ablation studies\n\n\nweaknesses:\n- simplicial complex computation is compute heavy\n- dataset is limited to biological data and makes it harder to judge the generalizability aspect of HiPoNet\n- background information about the differentiable Vietoris-Rips complex is scarce", "was_revised": true, "final_justification": "After reading responses to my questions and other reviewers questions, I will keep my current rating.", "rebuttal": "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:\n\n### **Custom differentiable Vietoris-Rips implementation**\n\nWe 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.\n\n### **Computational Complexity of Simplicial Complexes**\n\nWe 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.\n\n### **Application to non-biological datasets**\n\nWhile 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.\n\n### **Clustering and Logistic Regression Baselines**\n\nWe 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.\n\n### **Optimal number of views**\n\nThank 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.\n\n### **Hyperparameter selection**\n\nFor 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.\n\n**Overall:** \n\nWe 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\\!\n\n\\[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."}, {"review_id": "mdQPkCDQkQ", "reviewer": "Reviewer_cvsy", "summary": "The paper introduces a novel neural network architecture specifically designed for analyzing high-dimensional point clouds, particularly from single-cell and spatial transcriptomics data. Contributions include: 1. HiPoNet learns multiple weighted views of high-dimensional data. 2. HiPoNet uses higher-order simplicial complexes to model data.", "questions": "1. Can authors explain computational bottlenecks of HiPoNet in terms of memory and runtime for large-scale datasets, especially when using multiple views and higher-order simplices?\n2. Can authors explain how robust is HiPoNet to changes in the Vietoris–Rips threshold, kernel bandwidth, and number of views?\n3. Can authors explain the possibility of applying this method to non-biological high-dimensional point clouds?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. It models high-dimensional point clouds using simplicial complexes rather than standard graphs.\n2. The learnable feature reweighting mechanism allows the model to generate multiple views of the data.\n3. This paper outperform other methods across different real-world datasets. \n\nWeaknesses:\n1. The architecture becomes computationally intensive when processing multiple learned views.\n2. The model is heavily specialized to biological data, which may not be suitable for other types of data.", "was_revised": false, "final_justification": "", "rebuttal": "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:\n\n### **Computational Complexity of HiPoNet**\n\nWe 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.\n\n### **Sensitivity to Hyperparameters**\n\nTo 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.\n\nTo 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.\n\nIn 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.\n\n### **Application to non-Biological data**\n\nWhile 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.\n\n**Overall:** \n\nWe 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\\!"}, {"review_id": "ZXFAw4Avb4", "reviewer": "Reviewer_rx7G", "summary": "This paper presents a methodology, HiPoNet, for analysing high-dimensional point clouds. This proceeds by constructing several different Vietoris-Rips complexes that capture different aspects of the data, then vectorising these simplicial complexes using simplicial wavelets. The vectorisations of the simplicial complexes are then concatenated and used as input to a multilayer perceptron, with the entire stack being trained (end-to-end) on the appropriate classification problem for that point cloud.\n\nThe authors then apply their methodology to three datasets of medical data that feature large numbers of datapoints per patient, with each datapoint being information about a cell from that patient.", "questions": "Is your definition of upper adjacent neighbours correct? The definition in words does not match the definition in mathematical symbols unless every pair of k-simplices that meets along a (k-1)-simplex together span a (k+1)-simplex, which is quite close to the notion of the simplicial complex being a Kan complex. (Also (k+1)-simplices have k+2 vertices, so there is a typo in the formula there.)\n\nCan you analyse or give a conceptual justification for the definition of $P_k$ in the oriented case, just below equation 3? Also, in the non-oriented case, are you working with integer coefficients or with coefficients in the integers modulo 2? The latter would be more conventional, as otherwise e.g. the boundary operator won't square to zero in general.\n\nIn algorithm 1, on lines 7 and 8, you construct features by successively applying boundary operators. But doesn't the boundary operator square to zero? That is: why isn't $B_k^T B_{k-1}^T$ zero?\n\nIn the section, \"Interpretability of Learned Features\", currently your logic goes the wrong way around: you identify that known-meaningful biomarkers have high importance across views. Are there other features that have high importance across views? Can you determine whether or not these are known to be meaningful?", "limitations": "There is no limitations section. I assume that this is in the Appendices, which were not provided.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n* The empirical performance of HiPoNet described in Section 5 is the strongest part of the paper\n\n## Weaknesses\n\n* The theoretical justifications presented throughout the paper are weak, and basic definitions are often either confusing or confused. See the questions below for some examples.\n\n* The choice to apply the absolute value to all of the wavelet transforms, just after equation 4, lacks conceptual justification.\n\n* The theoretical analysis in Section 4 lacks detail and precision -- it is vague and sketchy --  and is all about heat flow on simplicial complexes whereas the algorithm presented does not involve heat flow at all. The purported justification for analysing heat flow is that the algorithm uses random walks, which are proxies for the diffusion of heat, but given the way the \"random walks\" are calculated (see below), it is not clear, at least to me, that this link is valid.\n\n* In the definition of \"random walk matrices\" $P_k$  just below equation 3, the authors choose to ignore all signs, even in the oriented case. This is likely to make a very big difference, and requires analysis and/or conceptual justification.", "was_revised": true, "final_justification": "Updated in light of the discussion with authors and the responses to other reviewers..", "rebuttal": "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:\n\n### **Regarding missing Appendices**\n\nAs 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. \n\n### **Applying Absolute Value**\n\nOne 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.\n\n### **Connection between Heat flow and Diffusion**\n\nThe 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.\n\n### **Use of Random Walks on Oriented Simplicial Complex**\n\nIn 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.\n\nThe 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\\].\n\n### **Use of Integer Coefficients in the Non-Oriented Case and Implications for Topological Properties**\n\nFor 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. \n\n### **HiPoNet Captures Known Biology**\n\nWhen 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.\n\nWe 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\\!\n\n\\[1\\] Mallat, Stéphane. \"Group invariant scattering.\" *Communications on Pure and Applied Mathematics* 65.10 (2012): 1331-1398.\n\n\\[2\\] Gama, Fernando, Alejandro Ribeiro, and Joan Bruna. \"Diffusion scattering transforms on graphs.\" *arXiv preprint arXiv:1806.08829* (2018).\n\n\\[3\\] Gao, Feng, Guy Wolf, and Matthew Hirn. \"Geometric scattering for graph data analysis.\" *International Conference on Machine Learning*. PMLR, 2019\\.\n\n\\[4\\] Chew, Joyce, et al. \"Geometric scattering on measure spaces.\" *Applied and Computational Harmonic Analysis* 70 (2024): 101635\\.\n\n\\[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\\.\n\n\\[6\\] Huguet, Guillaume, et al. \"A heat diffusion perspective on geodesic preserving dimensionality reduction.\" *Advances in Neural Information Processing Systems* 36 (2023): 6986-7016.\n\n\\[7\\] Jones, Iolo. \"Diffusion geometry.\" *arXiv preprint arXiv:2405.10858* (2024).\n\n\\[8\\] Coifman, Ronald R., and Stéphane Lafon. \"Diffusion maps.\" *Applied and computational harmonic analysis* 21.1 (2006): 5-30.\n\n\\[9\\] Coifman, Ronald R., and Mauro Maggioni. \"Diffusion wavelets.\" *Applied and computational harmonic analysis* 21.1 (2006): 53-94.\n\n\\[10\\] Chew, Joyce, et al. \"Manifold Filter-Combine Networks.\" *arXiv preprint arXiv:2307.04056* (2023)."}]

Author rebuttal:

Meta-review: HiPoNet introduces a framework for analyzing high-dimensional point clouds using multi-view simplicial complexes and wavelet transforms, with a strong focus on biological datasets such as single-cell and spatial transcriptomics data. 

The reviewers found the method well-motivated, empirically strong, and demonstrates significant performance gains over existing relevant baselines particularly in clinically relevant tasks. Reviewers appreciated the biological relevance, multi-view design, and ablation studies, while initial concerns around theoretical clarity and computational complexity were addressed sufficiently well in the rebuttal. Reviewers generally acknowledged the authors’ clarifications, and multiple scores were raised during discussion. 

Overall, HiPoNet represents a technically solid and potentially impactful contribution to geometric deep learning and scientific machine learning. I therefore recommend acceptance with the expectation that the additional results and clarification that emerged during the discussion phase are included in the final version of the paper.
Decision: Accept (poster)