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Register a new userA Simple Proof of the Universality of Invariant/Equivariant Graph Neural Networks
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We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality conclusions including an open case for higher-order output. We find that our proposed technique not only leads to simple proofs of the universality properties but also gives a natural explanation for the tensorization of the previously studied models. Finally, we give some remarks on the connection between our model and the continuous representation of graphs.
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This paper builds on the connection between graph neural networks and traditional dynamical systems. We propose continuous graph neural networks (CGNN), which generalise existing graph neural networks with discrete dynamics in that they can be viewed as a specific discretisation scheme. The key idea is how to characterise the continuous dynamics of node representations, i.e. the derivatives of node representations, w.r.t. time. Inspired by existing diffusion-based methods on graphs (e.g. PageRank and epidemic models on social networks), we define the derivatives as a combination of the current node representations, the representations of neighbors, and the initial values of the nodes. We propose and analyse two possible dynamics on graphs---including each dimension of node representations (a.k.a. the feature channel) change independently or interact with each other---both with theoretical justification. The proposed continuous graph neural networks are robust to over-smoothing and hence allow us to build deeper networks, which in turn are able to capture the long-range dependencies between nodes. Experimental results on the task of node classification demonstrate the effectiveness of our proposed approach over competitive baselines.
Graph Neural Networks (GNNs) are widely used deep learning models that learn meaningful representations from graph-structured data. Due to the finite nature of the underlying recurrent structure, current GNN methods may struggle to capture long-range dependencies in underlying graphs. To overcome this difficulty, we propose a graph learning framework, called Implicit Graph Neural Networks (IGNN), where predictions are based on the solution of a fixed-point equilibrium equation involving implicitly defined state vectors. We use the Perron-Frobenius theory to derive sufficient conditions that ensure well-posedness of the framework. Leveraging implicit differentiation, we derive a tractable projected gradient descent method to train the framework. Experiments on a comprehensive range of tasks show that IGNNs consistently capture long-range dependencies and outperform the state-of-the-art GNN models.
Learning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant to all three shape-preserving transformations of point clouds: translation, rotation, and permutation. In this paper, we present a first study of the approximation power of these architectures. We first derive two sufficient conditions for an equivariant architecture to have the universal approximation property, based on a novel characterization of the space of equivariant polynomials. We then use these conditions to show that two recently suggested models are universal, and for devising two other novel universal architectures.
Graph neural networks (GNN) has been successfully applied to operate on the graph-structured data. Given a specific scenario, rich human expertise and tremendous laborious trials are usually required to identify a suitable GNN architecture. It is because the performance of a GNN architecture is significantly affected by the choice of graph convolution components, such as aggregate function and hidden dimension. Neural architecture search (NAS) has shown its potential in discovering effective deep architectures for learning tasks in image and language modeling. However, existing NAS algorithms cannot be directly applied to the GNN search problem. First, the search space of GNN is different from the ones in existing NAS work. Second, the representation learning capacity of GNN architecture changes obviously with slight architecture modifications. It affects the search efficiency of traditional search methods. Third, widely used techniques in NAS such as parameter sharing might become unstable in GNN. To bridge the gap, we propose the automated graph neural networks (AGNN) framework, which aims to find an optimal GNN architecture within a predefined search space. A reinforcement learning based controller is designed to greedily validate architectures via small steps. AGNN has a novel parameter sharing strategy that enables homogeneous architectures to share parameters, based on a carefully-designed homogeneity definition. Experiments on real-world benchmark datasets demonstrate that the GNN architecture identified by AGNN achieves the best performance, comparing with existing handcrafted models and tradistional search methods.
The graph Laplacian regularization term is usually used in semi-supervised representation learning to provide graph structure information for a model $f(X)$. However, with the recent popularity of graph neural networks (GNNs), directly encoding graph structure $A$ into a model, i.e., $f(A, X)$, has become the more common approach. While we show that graph Laplacian regularization brings little-to-no benefit to existing GNNs, and propose a simple but non-trivial variant of graph Laplacian regularization, called Propagation-regularization (P-reg), to boost the performance of existing GNN models. We provide formal analyses to show that P-reg not only infuses extra information (that is not captured by the traditional graph Laplacian regularization) into GNNs, but also has the capacity equivalent to an infinite-depth graph convolutional network. We demonstrate that P-reg can effectively boost the performance of existing GNN models on both node-level and graph-level tasks across many different datasets.
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