  Pairwise comparison of graphs is key to many applications in Machine learning
ranging from clustering, kernel-based classification/regression and more
recently supervised graph prediction. Distances between graphs usually rely on
informative representations of these structured objects such as bag of
substructures or other graph embeddings. A recently popular solution consists
in representing graphs as metric measure spaces, allowing to successfully
leverage Optimal Transport, which provides meaningful distances allowing to
compare them: the Gromov-Wasserstein distances. However, this family of
distances overlooks edge attributes, which are essential for many structured
objects. In this work, we introduce an extension of Gromov-Wasserstein distance
for comparing graphs whose both nodes and edges have features. We propose novel
algorithms for distance and barycenter computation. We empirically show the
effectiveness of the novel distance in learning tasks where graphs occur in
either input space or output space, such as classification and graph
prediction.
