Clustering with Total Variation Graph Neural Networks

Graph Neural Networks (GNNs) are deep learning models designed to process attributed graphs. GNNs can compute cluster assignments accounting both for the vertex features and for the graph topology. Existing GNNs for clustering are trained by optimizing an unsupervised minimum cut objective, which is approximated by a Spectral Clustering (SC) relaxation. SC offers a closed-form solution that, however, is not particularly useful for a GNN trained with gradient descent. Additionally, the SC relaxation is loose and yields overly smooth cluster assignments, which do not separate well the samples. We propose a GNN model that optimizes a tighter relaxation of the minimum cut based on graph total variation (GTV). Our model has two core components: i) a message-passing layer that minimizes the $\ell_1$ distance in the features of adjacent vertices, which is key to achieving sharp cluster transitions; ii) a loss function that minimizes the GTV in the cluster assignments while ensuring balanced partitions. By optimizing the proposed loss, our model can be self-trained to perform clustering. In addition, our clustering procedure can be used to implement graph pooling in deep GNN architectures for graph classification. Experiments show that our model outperforms other GNN-based approaches for clustering and graph pooling.