In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Single-cell data can have high dimensionality exceeding the capabilities of existing methods point cloud tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e. one on 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 information. In contrast, HiPoNet forms higher-order simplicial complexes through learnable feature reweighting, generating multiple data views that disentangle distinct biological processes. It then employs simplicial wavelet transforms to extract multi-scale features - capturing both local and global topology. We empirically show that these components preserve topological information in the learned representations, and that HiPoNet significantly outperforms state-of-the-art point-cloud and graph-based models on single cell. We also show an application of HiPoNet on spatial transcriptomics datasets using spatial co-ordinates as one of the views. Overall, HiPoNet offers a robust and scalable solution for high-dimensional data analysis.

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity.

Graph neural networks (GNNs) have achieved great success for a variety of tasks such as node classification, graph classification, and link prediction. However, the use of GNNs (and machine learning more generally) to solve combinatorial optimization (CO) problems is much less explored. Here, we introduce a novel GNN architecture which leverages a complex filter bank and localized attention mechanisms designed to solve CO problems on graphs. We show how our method differentiates itself from prior GNN-based CO solvers and how it can be effectively applied to the maximum clique, minimum dominating set, and maximum cut problems in a self-supervised learning setting. In addition to demonstrating competitive overall performance across all tasks, we establish state-of-the-art results for the max cut problem.

Here we consider the problem of denoising features associated to complex data, modeled as signals on a graph, via a smoothness prior. This is motivated in part by settings such as single-cell RNA where the data is very high-dimensional, but its structure can be captured via an affinity graph. This allows us to utilize ideas from graph signal processing. In particular, we present algorithms for the cases where the signal is perturbed by Gaussian noise, dropout, and uniformly distributed noise. The signals are assumed to follow a prior distribution defined in the frequency domain which favors signals which are smooth across the edges of the graph. By pairing this prior distribution with our three models of noise generation, we propose Maximum A Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data and provide algorithms for computing the M.A.P. Finally, we demonstrate the algorithms' ability to effectively restore signals from white noise on image data and from severe dropout in single-cell RNA sequence data.

In this paper, we propose Graph Differential Equation Network (GDeNet), an approach that harnesses the expressive power of solutions to PDEs on a graph to obtain continuous node- and graph-level representations for various downstream tasks. We derive theoretical results connecting the dynamics of heat and wave equations to the spectral properties of the graph and to the behavior of continuous-time random walks on graphs. We demonstrate experimentally that these dynamics are able to capture salient aspects of graph geometry and topology by recovering generating parameters of random graphs, Ricci curvature, and persistent homology. Furthermore, we demonstrate the superior performance of GDeNet on real-world datasets including citation graphs, drug-like molecules, and proteins.

Directed graphs are a natural model for many phenomena, in particular scientific knowledge graphs such as molecular interaction or chemical reaction networks that define cellular signaling relationships. In these situations, source nodes typically have distinct biophysical properties from sinks. Due to their ordered and unidirectional relationships, many such networks also have hierarchical and multiscale structure. However, the majority of methods performing node- and edge-level tasks in machine learning do not take these properties into account, and thus have not been leveraged effectively for scientific tasks such as cellular signaling network inference. We propose a new framework called Directed Scattering Autoencoder (DSAE) which uses a directed version of a geometric scattering transform, combined with the non-linear dimensionality reduction properties of an autoencoder and the geometric properties of the hyperbolic space to learn latent hierarchies. We show this method outperforms numerous others on tasks such as embedding directed graphs and learning cellular signaling networks.

We introduce a class of manifold neural networks (MNNs) that we call Manifold Filter-Combine Networks (MFCNs), which aims to further our understanding of MNNs, analogous to how the aggregate-combine framework helps with the understanding of graph neural networks (GNNs). This class includes a wide variety of subclasses that can be thought of as the manifold analog of various popular GNNs. We then consider a method, based on building a data-driven graph, for implementing such networks when one does not have global knowledge of the manifold, but merely has access to finitely many points sampled from some probability distribution on the manifold. We provide sufficient conditions for the network to provably converge to its continuum limit as the number of sample points tends to infinity. Unlike previous work (which focused on specific graph constructions and assumed that the data was drawn from the uniform distribution), our rate of convergence does not directly depend on the number of filters used. Moreover, it exhibits linear dependence on the depth of the network rather than the exponential dependence obtained previously. Additionally, we provide several examples of interesting subclasses of MFCNs and of the rates of convergence that are obtained under specific graph constructions.

We consider the problem of embedding point cloud data sampled from an underlying manifold with an associated flow or velocity. Such data arises in many contexts where static snapshots of dynamic entities are measured, including in high-throughput biology such as single-cell transcriptomics. Existing embedding techniques either do not utilize velocity information or embed the coordinates and velocities independently, i.e., they either impose velocities on top of an existing point embedding or embed points within a prescribed vector field. Here we present FlowArtist, a neural network that embeds points while jointly learning a vector field around the points. The combination allows FlowArtist to better separate and visualize velocity-informed structures. Our results, on toy datasets and single-cell RNA velocity data, illustrate the value of utilizing coordinate and velocity information in tandem for embedding and visualizing high-dimensional data.

High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.

Graph Neural Networks (GNNs) extend the success of neural networks to graph-structured data by accounting for their intrinsic geometry. While extensive research has been done on developing GNN models with superior performance according to a collection of graph representation learning benchmarks, it is currently not well understood what aspects of a given model are probed by them. For example, to what extent do they test the ability of a model to leverage graph structure vs. node features? Here, we develop a principled approach to taxonomize benchmarking datasets according to a $\textit{sensitivity profile}$ that is based on how much GNN performance changes due to a collection of graph perturbations. Our data-driven analysis provides a deeper understanding of which benchmarking data characteristics are leveraged by GNNs. Consequently, our taxonomy can aid in selection and development of adequate graph benchmarks, and better informed evaluation of future GNN methods. Finally, our approach and implementation in $\texttt{GTaxoGym}$ package are extendable to multiple graph prediction task types and future datasets.