HesGCN: Hessian graph convolutional networks for semi-supervised classification

Manifold or local geometry of samples have been recognized as a powerful tool in machine learning areas, especially in the graph-based semi-supervised learning (GSSL) problems. Over recent decades, plenty of manifold assumption-based SSL algorithms (MSSL) have been proposed including graph embedding and graph regularization models, where the objective is to utilize the local geometry of data distributions. One of most representative MSSL approaches is graph convolutional networks (GCN), which effectively generalizes the convolutional neural networks to deal with the graphs with the arbitrary structures by constructing and fusing the Laplacian-based structure information. However, the null space of the Laplacian remains unchanged along the underlying manifold, it causes the poor extrapolating ability of the model. In this paper, we introduce a variant of GCN, i.e. Hessian graph convolutional networks (HesGCN). In particularly, we get a more efficient convolution layer rule by optimizing the one-order spectral graph Hessian convolutions. In addition, the spectral graph Hessian convolutions is a combination of the Hessian matrix and the spectral graph convolutions. Hessian gets a richer null space by the existence of its two-order derivatives, which can describe the intrinsic local geometry structure of data accurately. Thus, HesGCN can learn more efficient data features by fusing the original feature information with its structure information based on Hessian. We conduct abundant experiments on four public datasets. Extensive experiment results validate the superiority of our proposed HesGCN compared with many state-of-the-art methods.