Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport

Hierarchical abstractions are a methodology for solving large-scale graph problems in various disciplines. Coarsening is one such approach: it generates a pyramid of graphs whereby the one in the next level is a structural summary of the prior one. With a long history in scientific computing, many coarsening strategies were developed based on mathematically driven heuristics. Recently, resurgent interests exist in deep learning to design hierarchical methods learnable through differentiable parameterization. These approaches are paired with downstream tasks for supervised learning. In practice, however, supervised signals (e.g., labels) are scarce and are often laborious to obtain. In this work, we propose an unsupervised approach, coined OTCOARSEN-ING, with the use of optimal transport. Both the coarsening matrix and the transport cost matrix are parameterized, so that an optimal coarsening strategy can be learned and tailored for a given set of graphs without use of labels. We demonstrate that the proposed approach produces meaningful coarse graphs and yields competitive performance compared with supervised methods for graph classification and regression.