# Geometric Scattering on Measure Spaces arXiv: https://arxiv.org/abs/2208.08561 ยท HTML: https://mlsanity.com/paper/2208.08561/ ## Key facts - Cited by 17 papers in the map (data through 14 Oct 2026). - Builds on 44 of its 81 references that are in the map. - First posted on arXiv 17 Aug 2022. ## Builds on (most cited first, 30 of 44) - [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016) - [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017) - [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017) - [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018) - [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016) - [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016) - [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013) - [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016) - [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018) - [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018) - [graph2vec: Learning Distributed Representations of Graphs](https://mlsanity.com/paper/1707.05005/) (2017) - [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017) - [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021) - [Learning shape correspondence with anisotropic convolutional neural networks](https://mlsanity.com/paper/1605.06437/) (2016) - [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021) - [Directed Graph Convolutional Network](https://mlsanity.com/paper/2004.13970/) (2020) - [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022) - [Geometric Scattering for Graph Data Analysis](https://mlsanity.com/paper/1810.03068/) (2018) - [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015) - [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018) - [Diffusion Scattering Transforms on Graphs](https://mlsanity.com/paper/1806.08829/) (2018) - [Spectral-based Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/1907.08990/) (2019) - [Magnetic Eigenmaps for the Visualization of Directed Networks](https://mlsanity.com/paper/1606.08266/) (2016) - [Can Hybrid Geometric Scattering Networks Help Solve the Maximum Clique Problem?](https://mlsanity.com/paper/2206.01506/) (2022) - [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021) - [SigMaNet: One Laplacian to Rule Them All](https://mlsanity.com/paper/2205.13459/) (2022) - [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016) - [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021) - [Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networks](https://mlsanity.com/paper/2201.08932/) (2022) - [Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms](https://mlsanity.com/paper/1712.06206/) (2017) ## Built on by (most cited first, 17 of 17) - [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024) - [A Convergence Rate for Manifold Neural Networks](https://mlsanity.com/paper/2212.12606/) (2022) - [DYMAG: Rethinking Message Passing Using Dynamical-systems-based Waveforms](https://mlsanity.com/paper/2309.09924/) (2023) - [BLIS-Net: Classifying and Analyzing Signals on Graphs](https://mlsanity.com/paper/2310.17579/) (2023) - [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023) - [Directed Scattering for Knowledge Graph-based Cellular Signaling Analysis](https://mlsanity.com/paper/2309.07813/) (2023) - [Wavelet Scattering Transform and Fourier Representation for Offline Detection of Malicious Clients in Federated Learning](https://mlsanity.com/paper/2506.09674/) (2025) - [Multiscale Hodge Scattering Networks for Data Analysis](https://mlsanity.com/paper/2311.10270/) (2023) - [Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slow](https://mlsanity.com/paper/2406.05121/) (2024) - [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025) - [Convergence of Manifold Filter-Combine Networks](https://mlsanity.com/paper/2410.14639/) (2024) - [Digital implementations of deep feature extractors are intrinsically informative](https://mlsanity.com/paper/2502.15004/) (2025) - [InfoGain Wavelets: Furthering the Design of Graph Diffusion Wavelets](https://mlsanity.com/paper/2504.08802/) (2025) - [Scattering Networks on Noncommutative Finite Groups](https://mlsanity.com/paper/2505.20950/) (2025) - [SlepNet: Spectral Subgraph Representation Learning for Neural Dynamics](https://mlsanity.com/paper/2506.16602/) (2025) - [Can Computational Reducibility Lead to Transferable Models for Graph Combinatorial Optimization?](https://mlsanity.com/paper/2603.02462/) (2026) - [Transversal Pooling Neural Networks](https://mlsanity.com/paper/2609.36237/) (2026) ## Abstract (arXiv) The scattering transform is a multilayered, wavelet-based transform initially introduced as a model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. Subsequently, 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. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on geometric scattering 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, directed graphs, and on high-dimensional single-cell data.