Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs

By Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodolà, Jan Svoboda, Michael M. Bronstein. 2016

1,972 citations recorded by the source corpus.

Deep learning has achieved a remarkable performance breakthrough in several fields, most notably in speech recognition, natural language processing, and computer vision. In particular, convolutional neural network (CNN) architectures currently produce state-of-the-art performance on a variety of image analysis tasks such as object detection and recognition. Most of deep learning research has so far focused on dealing with 1D, 2D, or 3D Euclidean-structured data such as acoustic signals, images, or videos. Recently, there has been an increasing interest in geometric deep learning, attempting to generalize deep learning methods to non-Euclidean structured data such as graphs and manifolds, with a variety of applications from the domains of network analysis, computational social science, or computer graphics. In this paper, we propose a unified framework allowing to generalize CNN architectures to non-Euclidean domains (graphs and manifolds) and learn local, stationary, and compositional task-specific features. We show that various non-Euclidean CNN methods previously proposed in the literature can be considered as particular instances of our framework. We test the proposed method on standard tasks from the realms of image-, graph-and 3D shape analysis and show that it consistently outperforms previous approaches.

Subfields: Artificial Intelligence, Computational Mechanics, Computer Vision and Pattern Recognition, Model Architectures, Graph Learning

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Citations in the corpus

Graph Attention Networks (2017, methodology): “More speciﬁcally, setting the pseudo-coordinate function to be u ( x, y ) = f ( x ) k f ( y ) , where f ( x ) represent (potentially MLP-transformed) features of node x and k is concatenation; and the weight function to be w j ( u ) = softmax(MLP( u )) (with the softmax performed over the entire nei”
Fast Graph Representation Learning with PyTorch Geometric (2019, methodology): “In addition, we provide embedded datasets like MNIST superpixels (Monti et al., 2017), FAUST (Bogo et al., 2014), ModelNet10/40 (Wu et al., 2015), ShapeNet (Chang et al., 2015), COMA (Ranjan et al., 2018), and the PCPNet dataset from Guerrero et al. (2018).”
Relational inductive biases, deep learning, and graph networks (2018, background): “(2017) introduced the message-passing neural network (MPNN), which unified various graph neural network and graph convolutional network approaches (Monti et al., 2017; Bruna et al., 2014; Henaff et al., 2015; Defferrard et al., 2016; Kipf and Welling, 2017; Bronstein et al., 2017) by analogy to mess”
Differentiable Physics-informed Graph Networks (2019, background): “One branch of these studies is a spectral approach to aggregate locally connected attributes (Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017) and another branch is a spatial approach (or message passing method) (Gilmer et al., 2017; Monti et al., 2017; Hamilton et al., 2017).”
Representation Learning on Graphs with Jumping Knowledge Networks (2018, background): “…from a node as well as its neighborhood, and has proved extremely useful for many applications, such as node classiﬁcation, clustering, and link prediction (Perozzi et al., 2014; Monti et al., Proceedings of the 35 th International Conference on Machine Learning , Stockholm, Sweden, PMLR 80, 2018.”
Junction Tree Variational Autoencoder for Molecular Graph Generation (2018, methodology): “For applications, graph neural networks are used in semi-supervised classiﬁcation (Kipf & Welling, 2016), computer vision (Monti et al., 2016), and chemical domains (Kearnes et al., 2016; Sch ¨ utt et al., 2017; Jin et al., 2017).”
Cormorant: Covariant Molecular Neural Networks (2019, background): “Similar ideas underly generalizations of CNNs to manifolds [Masci et al., 2015, Monti et al., 2016, Bronstein et al., 2017] and graphs [Bruna et al., 2014, Henaff et al., 2015].”
Provably Powerful Graph Networks (2019, background): “…deep models that are able to learn new graph representations from raw features (e.g., Gori et al. (2005); Duvenaud et al. (2015); Niepert et al. (2016); Kipf and Welling (2016); Veliˇckovi´c et al. (2017); Monti et al. (2017); Hamilton et al. (2017a); Morris et al. (2018); Xu et al. (2019)).”
Predict then Propagate: Graph Neural Networks meet Personalized PageRank (2018, background): “…2016), message passing (or neighbor aggregation) algorithms (Kearnes et al., 2016; Kipf & Welling, 2017; Hamilton et al., 2017; Pham et al., 2017; Monti et al., 2017; Gilmer et al., 2017), and neighbor aggregation via recurrent neural networks (Scarselli et al., 2009; Li et al., 2016; Dai et…”
Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network (2018, background): “The starting point for our Generalized Spherical CNNs is the Fourier space correlation formula (10).”
FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling (2018, background): “…a view, yet, generalizes to many graph models whose formulations are based on ﬁrst-order neighborhoods, examples of which include MoNet that applies to (meshed) manifolds (Monti et al., 2017), as well as many message-passing neural networks (see e.g., Scarselli et al. (2009); Gilmer et al. (2017)).”
Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks (2017, background): “…deep learning (Bron-stein et al., 2016) techniques that can mathematically deal with graph-structured data, which arises in numerous applications, ranging from computer graphics and vision (Masci et al., 2015; Boscaini et al., 2015; 2016b;a; Monti et al., 2017) to chemistry (Duvenaud et al., 2015).”
Geometric Deep Learning: Going beyond Euclidean data (2016, methodology): “[FIGS7b] Intrinsic correspondence established between human shapes using intrinsic deep architecture (MoNet [54] with three convolutional”
A Comprehensive Survey of Graph Embedding: Problems, Techniques, and Applications (2017, methodology): “, [80], [81], [82]) construct a graph from each image by treating pixels as nodes and the spatial relations between pixels as edges.”
Just Jump: Dynamic Neighborhood Aggregation in Graph Neural Networks (2019, background): “Many different graph neural network variants have been proposed and significantly advanced the state-of-the-art in this field (Defferrard et al., 2016; Kipf & Welling, 2017; Monti et al., 2017; Gilmer et al., 2017; Hamilton et al., 2017; Veličković et al., 2018; Fey et al., 2018).”
Simplifying Graph Convolutional Networks (2019, background): “There are many other graph neural models (Monti et al., 2017; Duran & Niepert, 2017; Li et al., 2018); we refer to Zhou et al.”
Pitfalls of Graph Neural Network Evaluation (2018, methodology): “We implement the four models – GCN [Kipf and Welling, 2017], MoNet [Monti et al., 2017], GraphSage [Hamilton et al., 2017] and GAT [Velickovic et al., 2018] – within the same framework.”
Towards Sparse Hierarchical Graph Classifiers (2018, background): “…broadly fall into two categories: 1) aggregating node representations in a global pooling step after each [8] or after the ﬁnal message passing step [19, 5, 11], and 2) aggregating node representations into clusters which coarsen the graph in a hierarchical manner [4, 22, 6, 20, 24, 10, 21, 28, 1].”
Generating 3D faces using Convolutional Mesh Autoencoders (2018, methodology): “[32] introduce d-dimensional pseudo-coordinates that define a local system around each point with weight functions.”
Attention-based Graph Neural Network for Semi-supervised Learning (2018, methodology): “The classiﬁcation accuracy of all the baseline methods are collected from [46, 27, 38, 31, 11, 41].”