Title: A Unified Framework for Convolution-based Graph Neural Networks. Abstract: Graph Convolutional Networks (GCNs) have attracted a lot of research interest in the machine learning community in recent years. Although many variants have been proposed, we still lack a systematic view of different GCN models and deep understanding of the relations among them. In this paper, we take a step forward to establish a unified framework for convolution-based graph neural networks, by formulating the basic graph convolution operation as an optimization problem in the graph Fourier space. Under this framework, a variety of popular GCN models, including the vanilla-GCNs, attention-based GCNs and topology-based GCNs, can be interpreted as a same optimization problem but with different carefully designed regularizers. This novel perspective enables a better understanding of the similarities and differences among many widely used GCNs, and may inspire new approaches for designing better models. As a showcase, we also present a novel regularization technique under the proposed framework to tackle the oversmoothing problem in graph convolution. The effectiveness of the newly designed model is validated empirically.