Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be prone to topological attacks. Despite substantial efforts to search for new architectures, it still remains a challenge to improve performance in both benign and adversarial situations simultaneously. In this paper, we re-examine the fundamental building block of GCN---the Laplacian operator---and highlight some basic flaws in the spatial and spectral domains. As an alternative, we propose an operator based on graph powering, and prove that it enjoys a desirable property of "spectral separation." Based on the operator, we propose a robust learning paradigm, where the network is trained on a family of "'smoothed" graphs that span a spatial and spectral range for generalizability. We also use the new operator in replacement of the classical Laplacian to construct an architecture with improved spectral robustness, expressivity and interpretability. The enhanced performance and robustness are demonstrated in extensive experiments.
One limitation of using Laplacian in GCN is that the first layer only sees'' features of adjacent nodes. Several works have been proposed to address this issue @cite_11 @cite_5 @cite_15 . For instance, adjacency matrix and its powered versions are used in @cite_11 @cite_5 @cite_15 to construct GCN layers. However, simply powering the adjacency does not extract spectral gap and may even make the eigenspectrum more sensitive to perturbations. The architecture in @cite_15 also introduced additional weight matrices for neighbors at different distances. But this could substantially increase the risk of overfitting in the low-data regime and make the network vulnerable to adversarial attacks.
@cite_5 @cite_15 @cite_11
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Many interesting problems in machine learning are being revisited with new deep learning tools. For graph-based semisupervised learning, a recent important development is graph convolutional networks (GCNs), which nicely integrate local vertex features and graph topology in the convolutional layers. Although the GCN model compares favorably with other state-of-the-art methods, its mechanisms are not clear and it still requires a considerable amount of labeled data for validation and model selection. In this paper, we develop deeper insights into the GCN model and address its fundamental limits. First, we show that the graph convolution of the GCN model is actually a special form of Laplacian smoothing, which is the key reason why GCNs work, but it also brings potential concerns of over-smoothing with many convolutional layers. Second, to overcome the limits of the GCN model with shallow architectures, we propose both co-training and self-training approaches to train GCNs. Our approaches significantly improve GCNs in learning with very few labels, and exempt them from requiring additional labels for validation. Extensive experiments on benchmarks have verified our theory and proposals. Graph Convolutional Networks (GCNs) have become a crucial tool on learning representations of graph vertices. The main challenge of adapting GCNs on large-scale graphs is the scalability issue that it incurs heavy cost both in computation and memory due to the uncontrollable neighborhood expansion across layers. In this paper, we accelerate the training of GCNs through developing an adaptive layer-wise sampling method. By constructing the network layer by layer in a top-down passway, we sample the lower layer conditioned on the top one, where the sampled neighborhoods are shared by different parent nodes and the over expansion is avoided owing to the fixed-size sampling. More importantly, the proposed sampler is adaptive and applicable for explicit variance reduction, which in turn enhances the training of our method. Furthermore, we propose a novel and economical approach to promote the message passing over distant nodes by applying skip connections. Intensive experiments on several benchmarks verify the effectiveness of our method regarding the classification accuracy while enjoying faster convergence speed. We extend spectral methods to random graphs with skewed degree distributions through a degree based normalization closely connected to the normalized Laplacian. The normalization is based on intuition drawn from perturbation theory of random matrices, and has the effect of boosting the expectation of the random adjacency matrix without increasing the variances of its entries, leading to better perturbation bounds. The primary implication of this result lies in the realm of spectral analysis of random graphs with skewed degree distributions, such as the ubiquitous "power law graphs". Mihail and Papadimitriou (2002) argued that for randomly generated graphs satisfying a power law degree distribution, spectral analysis of the adjacency matrix simply produces the neighborhoods of the high degree nodes as its eigenvectors, and thus miss any embedded structure. We present a generalization of their model, incorporating latent structure, and prove that after applying our transformation, spectral analysis succeeds in recovering the latent structure with high probability. Abstract In this work, we propose a technique that utilizes a fully convolutional network (FCN) to localize image splicing attacks. We first evaluated a single-task FCN (SFCN) trained only on the surface label. Although the SFCN is shown to provide superior performance over existing methods, it still provides a coarse localization output in certain cases. Therefore, we propose the use of a multi-task FCN (MFCN) that utilizes two output branches for multi-task learning. One branch is used to learn the surface label, while the other branch is used to learn the edge or boundary of the spliced region. We trained the networks using the CASIA v2.0 dataset, and tested the trained models on the CASIA v1.0, Columbia Uncompressed, Carvalho, and the DARPA NIST Nimble Challenge 2016 SCI datasets. Experiments show that the SFCN and MFCN outperform existing splicing localization algorithms, and that the MFCN can achieve finer localization than the SFCN.