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Abstract In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our 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 manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then 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, and we numerically demonstrate its effectiveness on real-world and synthetic data sets. 

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. 

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. 

The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks’ stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non- Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data. 

The scattering transform is a multilayered wavelet-based architecture that acts as a model of convolutional neural networks. Recently, several works have generalized the scattering transform to graph-structured data. Our work builds on these constructions by introducing windowed and nonwindowed geometric scattering transforms for graphs based on two very general classes wavelets, which are in most cases based on asymmetric matrices. We show that these transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. Therefore, it helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties. 

We propose a new graph neural network (GNN) module, based on relaxations of recently proposed geometric scattering transforms, which consist of a cascade of graph wavelet filters. Our learnable geometric scattering (LEGS) module enables adaptive tuning of the wavelets to encourage band-pass features to emerge in learned representations. The incorporation of our LEGS-module in GNNs enables the learning of longer-range graph relations compared to many popular GNNs, which often rely on encoding graph structure via smoothness or similarity between neighbors. Further, its wavelet priors result in simplified architectures with significantly fewer learned parameters compared to competing GNNs. We demonstrate the predictive performance of LEGS-based networks on graph classification benchmarks, as well as the descriptive quality of their learned features in biochemical graph data exploration tasks. Our results show that LEGS-based networks match or outperforms popular GNNs, as well as the original geometric scattering construction, on many datasets, in particular in biochemical domains, while retaining certain mathematical properties of handcrafted (non-learned) geometric scattering.