  We propose a new nonlinear factorization model for graphs that are with
topological structures, and optionally, node attributes. This model is based on
a pseudometric called Gromov-Wasserstein (GW) discrepancy, which compares
graphs in a relational way. It estimates observed graphs as GW barycenters
constructed by a set of atoms with different weights. By minimizing the GW
discrepancy between each observed graph and its GW barycenter-based estimation,
we learn the atoms and their weights associated with the observed graphs. The
model achieves a novel and flexible factorization mechanism under GW
discrepancy, in which both the observed graphs and the learnable atoms can be
unaligned and with different sizes. We design an effective approximate
algorithm for learning this Gromov-Wasserstein factorization (GWF) model,
unrolling loopy computations as stacked modules and computing gradients with
backpropagation. The stacked modules can be with two different architectures,
which correspond to the proximal point algorithm (PPA) and Bregman alternating
direction method of multipliers (BADMM), respectively. Experiments show that
our model obtains encouraging results on clustering graphs.
