FlowPool: Pooling Graph Representations with Wasserstein Gradient Flows

In several machine learning tasks for graph structured data, the graphs under consideration may be composed of a varying number of nodes. Therefore, it is necessary to design pooling methods that aggregate the graph representations of varying size to representations of fixed size which can be used in downstream tasks, such as graph classification. Existing graph pooling methods offer no guarantee with regards to the similarity of a graph representation and its pooled version. In this work we address this limitation by proposing FlowPool, a pooling method that optimally preserves the statistics of a graph representation to its pooled counterpart by minimizing their Wasserstein distance. This is achieved by performing a Wasserstein gradient flow with respect to the pooled graph representation. We propose a versatile implementation of our method which can take into account the geometry of the representation space through any ground cost. This implementation relies on the computation of the gradient of the Wasserstein distance with recently proposed implicit differentiation schemes. Our pooling method is amenable to automatic differentiation and can be integrated in end-to-end deep learning architectures. Further, FlowPool is invariant to permutations and can therefore be combined with permutation equivariant feature extraction layers in GNNs in order to obtain predictions that are independent of the ordering of the nodes. Experimental results demonstrate that our method leads to an increase in performance compared to existing pooling methods when evaluated in graph classification tasks.