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Register a new userLorentzian Graph Convolutional Networks
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Added by
Yiding Zhang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Graph convolutional networks (GCNs) have received considerable research attention recently. Most GCNs learn the node representations in Euclidean geometry, but that could have a high distortion in the case of embedding graphs with scale-free or hierarchical structure. Recently, some GCNs are proposed to deal with this problem in non-Euclidean geometry, e.g., hyperbolic geometry. Although hyperbolic GCNs achieve promising performance, existing hyperbolic graph operations actually cannot rigorously follow the hyperbolic geometry, which may limit the ability of hyperbolic geometry and thus hurt the performance of hyperbolic GCNs. In this paper, we propose a novel hyperbolic GCN named Lorentzian graph convolutional network (LGCN), which rigorously guarantees the learned node features follow the hyperbolic geometry. Specifically, we rebuild the graph operations of hyperbolic GCNs with Lorentzian version, e.g., the feature transformation and non-linear activation. Also, an elegant neighborhood aggregation method is designed based on the centroid of Lorentzian distance. Moreover, we prove some proposed graph operations are equivalent in different types of hyperbolic geometry, which fundamentally indicates their correctness. Experiments on six datasets show that LGCN performs better than the state-of-the-art methods. LGCN has lower distortion to learn the representation of tree-likeness graphs compared with existing hyperbolic GCNs. We also find that the performance of some hyperbolic GCNs can be improved by simply replacing the graph operations with those we defined in this paper.
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Graphs have been widely adopted to denote structural connections between entities. The relations are in many cases heterogeneous, but entangled together and denoted merely as a single edge between a pair of nodes. For example, in a social network graph, users in different latent relationships like friends and colleagues, are usually connected via a bare edge that conceals such intrinsic connections. In this paper, we introduce a novel graph convolutional network (GCN), termed as factorizable graph convolutional network(FactorGCN), that explicitly disentangles such intertwined relations encoded in a graph. FactorGCN takes a simple graph as input, and disentangles it into several factorized graphs, each of which represents a latent and disentangled relation among nodes. The features of the nodes are then aggregated separately in each factorized latent space to produce disentangled features, which further leads to better performances for downstream tasks. We evaluate the proposed FactorGCN both qualitatively and quantitatively on the synthetic and real-world datasets, and demonstrate that it yields truly encouraging results in terms of both disentangling and feature aggregation. Code is publicly available at https://github.com/ihollywhy/FactorGCN.PyTorch.
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
Graph convolution networks have recently garnered a lot of attention for representation learning on non-Euclidean feature spaces. Recent research has focused on stacking multiple layers like in convolutional neural networks for the increased expressive power of graph convolution networks. However, simply stacking multiple graph convolution layers lead to issues like vanishing gradient, over-fitting and over-smoothing. Such problems are much less when using shallower networks, even though the shallow networks have lower expressive power. In this work, we propose a novel Multipath Graph convolutional neural network that aggregates the output of multiple different shallow networks. We train and test our model on various benchmarks datasets for the task of node property prediction. Results show that the proposed method not only attains increased test accuracy but also requires fewer training epochs to converge. The full implementation is available at https://github.com/rangan2510/MultiPathGCN
Graph Convolutional Networks (GCNs) are typically studied through the lens of Euclidean geometry. Non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or spherical data, but cannot align well with data of mixed topologies. We consider a larger class of semi-Riemannian manifolds with indefinite metric that generalize hyperboloid and sphere as well as their submanifolds. We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected semi-Riemannian manifolds. As a consequence, we derive a principled Semi-Riemannian GCN that first models data in semi-Riemannian manifolds of constant nonzero curvature in the context of graph neural networks. Our method provides a geometric inductive bias that is sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles. Empirical results demonstrate that our method outperforms Riemannian counterparts when embedding graphs of complex topologies.
Graph convolution operator of the GCN model is originally motivated from a localized first-order approximation of spectral graph convolutions. This work stands on a different view; establishing a textit{mathematical connection between graph convolution and graph-regularized PCA} (GPCA). Based on this connection, GCN architecture, shaped by stacking graph convolution layers, shares a close relationship with stacking GPCA. We empirically demonstrate that the textit{unsupervised} embeddings by GPCA paired with a 1- or 2-layer MLP achieves similar or even better performance than GCN on semi-supervised node classification tasks across five datasets including Open Graph Benchmark footnote{url{https://ogb.stanford.edu/}}. This suggests that the prowess of GCN is driven by graph based regularization. In addition, we extend GPCA to the (semi-)supervised setting and show that it is equivalent to GPCA on a graph extended with ghost edges between nodes of the same label. Finally, we capitalize on the discovered relationship to design an effective initialization strategy based on stacking GPCA, enabling GCN to converge faster and achieve robust performance at large number of layers. Notably, the proposed initialization is general-purpose and applies to other GNNs.
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