AnchorGAE: General Data Clustering via $O(n)$ Bipartite Graph Convolution

Since the representative capacity of graph-based clustering methods is usually limited by the graph constructed on the original features, it is attractive to find whether graph neural networks (GNNs) can be applied to augment the capacity. The core problems mainly come from two aspects: (1) the graph is unavailable in the most clustering scenes so that how to construct high-quality graphs on the non-graph data is usually the most important part; (2) given n samples, the graph-based clustering methods usually consume at least $\mathcal O(n^2)$ time to build graphs and the graph convolution requires nearly $\mathcal O(n^2)$ for a dense graph and $\mathcal O(|\mathcal{E}|)$ for a sparse one with $|\mathcal{E}|$ edges. Accordingly, both graph-based clustering and GNNs suffer from the severe inefficiency problem. To tackle these problems, we propose a novel clustering method, AnchorGAE, with the self-supervised estimation of graph and efficient graph convolution. We first show how to convert a non-graph dataset into a graph dataset, by introducing the generative graph model and anchors. We then show that the constructed bipartite graph can reduce the computational complexity of graph convolution from $\mathcal O(n^2)$ and $\mathcal O(|\mathcal{E}|)$ to $\mathcal O(n)$. The succeeding steps for clustering can be easily designed as $\mathcal O(n)$ operations. Interestingly, the anchors naturally lead to siamese architecture with the help of the Markov process. Furthermore, the estimated bipartite graph is updated dynamically according to the features extracted by GNN, to promote the quality of the graph. However, we theoretically prove that the self-supervised paradigm frequently results in a collapse that often occurs after 2-3 update iterations in experiments, especially when the model is well-trained. A specific strategy is accordingly designed to prevent the collapse.