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.
The robustness of spectral methods has been extensively studied for graph partitioning clustering using adversarial perturbations or noise perturbations @cite_53 @cite_2 @cite_1 . A common approach is to use regularization on the Laplacian @cite_54 @cite_24 @cite_16 . Most recently, operators based on self-avoiding or nonbacktracking walks have become popular for SBM @cite_51 @cite_30 @cite_28 , which have been proved to achieve the theoretical detection threshold conjectured in @cite_27 . Our work is partly motivated by the graph powering approach @cite_32 , which leveraged the result of @cite_51 @cite_28 to show a spectral gap of the powering operator. The main difference with this line of work is that these operators are studied only for spectral clustering algorithms, which does not incorporate nodal features. Our proposed variable power operator can be viewed as a kernel version of the graph powering operator @cite_32 , thus substantially increasing the capability of learning complex nodal interactions while maintaining the nice spectral property.
@cite_30 @cite_28 @cite_53 @cite_54 @cite_1 @cite_32 @cite_24 @cite_27 @cite_2 @cite_16 @cite_51
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Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix @math , one may look at the spectrum of @math for a properly chosen @math . The issue is that the spectrum of @math might be contaminated by non-informational top eigenvalues, e.g., due to scale variations in the data, and the application of @math aims to remove these. Designing a good functional @math (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, @math where @math denotes the adjacency matrix and @math is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance @math , and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd o s-R 'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to massoulie-STOC , settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts massoulie-STOC,Mossel_SBM2,bordenave,colin3 . This is illustrated on a geometric block model that is dense in cliques. In this paper, we consider the problem of unsupervised feature selection. Recently, spectral feature selection algorithms, which leverage both graph Laplacian and spectral regression, have received increasing attention. However, existing spectral feature selection algorithms suffer from two major problems: 1) since the graph Laplacian is constructed from the original feature space, noisy and irrelevant features may have adverse effect on the estimated graph Laplacian and hence degenerate the quality of the induced graph embedding, 2) since the cluster labels are discrete in natural, relaxing and approximating these labels into a continuous embedding can inevitably introduce noise into the estimated cluster labels. Without considering the noise in the cluster labels, the feature selection process may be misguided. In this paper, we propose a Robust Spectral learning framework for unsupervised Feature Selection (RSFS), which jointly improves the robustness of graph embedding and sparse spectral regression. Compared with existing methods which are sensitive to noisy features, our proposed method utilizes a robust local learning method to construct the graph Laplacian and a robust spectral regression method to handle the noise on the learned cluster labels. In order to solve the proposed optimization problem, an efficient iterative algorithm is proposed. We also show the close connection between the proposed robust spectral regression and robust Huber M-estimator. Experimental results on different datasets show the superiority of RSFS. Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Laplacian. We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices. One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to incorporate into existing implementations. We tested the efficacy of spectral normalization on CIFAR10, STL-10, and ILSVRC2012 dataset, and we experimentally confirmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques.