Hierarchical graph representation learning is an emerging subject owing to the increasingly popular adoption of graph neural networks in machine learning and applications. Loosely speaking, work under this umbrella falls into two categories: (a) use a predefined graph hierarchy to perform pooling; and (b) learn the hierarchy for a given graph through differentiable parameterization of the coarsening process. These approaches are supervised; a predictive task with ground-truth labels is used to drive the learning. In this work, we propose an unsupervised approach, BAYESPOOL, with the use of variational Bayes. It produces graph representations given a predefined hierarchy. Rather than relying on labels, the training signal comes from the evidence lower bound of encoding a graph and decoding the subsequent one in the hierarchy. Node features are treated latent in this variational machinery, so that they are produced as a byproduct and are used in downstream tasks. We demonstrate a comprehensive set of experiments to show the usefulness of the learned representation in the context of graph classification.
The authors propose in this paper a new unsupervised graph representation learning method. The method leverages recent advances in graph coarsening, mainly Loukas' method. The key idea of the method consists in using a reconstruction target that is not the classical one in an auto-encoder setting. More precisely, the encoder takes as an input the original adjacency matrix and node features but the decode only aims at reconstructing the coarse adjacency matrix (obtained via Loukas' method). 
The experimental evaluation is quite thorough and shows that the method performs quite well, especially considering it is unsupervised but is compared to supervised representation methods. It would be nice to include statistical tests to assess the significance of the differences in cases were accuracies are very close one to another. A missing part would be to explore the relevance of the learned representation for other tasks (i.e. to use a multi task data set). Of course as the representation is learned in an unsupervised way, one can argue that the current evaluation is already providing an answer.
Overall, I find the paper clear, but the variational bayes part could be much clearer. In fact I'm not sure why this is presented as variational bayes and not only variational. I do not see any prior distribution over parameters, for instance. I understand that the recent "tradition" in variational auto-encoder is to use this terminology, but as a (part time) bayesian, this is a bit annoying.