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The class of multi-relational graph convolutional networks (MRGCNs) is a recent extension of standard graph convolutional networks (GCNs) to handle heterogenous graphs with multiple types of relationships. MRGCNs have been shown to yield results superior than traditional GCNs in various machine learning tasks. The key idea is to introduce a new kind of convolution operated on tensors that can effectively exploit correlations exhibited in multiple relationships. The main objective of this paper is to analyze the algorithmic stability and generalization guarantees of MRGCNs to confirm the usefulness of MRGCNs. Our contributions are of three folds. First, we develop a matrix representation of various tensor operations underneath MRGCNs to simplify the analysis significantly. Next, we prove the uniform stability of MRGCNs and deduce the convergence of the generalization gap to support the usefulness of MRGCNs. The analysis sheds lights on the design of MRGCNs, for instance, how the data should be scaled to achieve the uniform stability of the learning process. Finally, we provide experimental results to demonstrate the stability results. [ABSTRACT FROM AUTHOR]

Deep neural networks have achieved great success in solving many machine learning and computer vision problems. In this paper, we propose a deep neural network called the Tucker network derived from the Tucker format and analyze its expressive power. The results demonstrate that the Tucker network has exponentially higher expressive power than the shallow network. In other words, a shallow network with an exponential width is required to realize the same score function as that computed by the Tucker network. Moreover, we discuss the expressive power between the hierarchical Tucker tensor network (HT network) and the proposed Tucker network. To generalize the Tucker network into a deep version, we combine the hierarchical Tucker format and Tucker format to propose a deep Tucker tensor decomposition. Its corresponding deep Tucker network is presented. Experiments are conducted on three datasets: MNIST, CIFAR-10 and CIFAR-100. The results experimentally validate the theoretical results and show that the Tucker network and deep Tucker network have better performance than the shallow network and HT network. [ABSTRACT FROM AUTHOR]