  Comparing structured objects such as graphs is a fundamental operation
involved in many learning tasks. To this end, the Gromov-Wasserstein (GW)
distance, based on Optimal Transport (OT), has proven to be successful in
handling the specific nature of the associated objects. More specifically,
through the nodes connectivity relations, GW operates on graphs, seen as
probability measures over specific spaces. At the core of OT is the idea of
conservation of mass, which imposes a coupling between all the nodes from the
two considered graphs. We argue in this paper that this property can be
detrimental for tasks such as graph dictionary or partition learning, and we
relax it by proposing a new semi-relaxed Gromov-Wasserstein divergence. Aside
from immediate computational benefits, we discuss its properties, and show that
it can lead to an efficient graph dictionary learning algorithm. We empirically
demonstrate its relevance for complex tasks on graphs such as partitioning,
clustering and completion.
