  In graph representation learning, it is important that the complex geometric
structure of the input graph, e.g. hidden relations among nodes, is well
captured in embedding space. However, standard Euclidean embedding spaces have
a limited capacity in representing graphs of varying structures. A promising
candidate for the faithful embedding of data with varying structure is product
manifolds of component spaces of different geometries (spherical, hyperbolic,
or euclidean). In this paper, we take a closer look at the structure of product
manifold embedding spaces and argue that each component space in a product
contributes differently to expressing structures in the input graph, hence
should be weighted accordingly. This is different from previous works which
consider the roles of different components equally. We then propose
WEIGHTED-PM, a data-driven method for learning embedding of heterogeneous
graphs in weighted product manifolds. Our method utilizes the topological
information of the input graph to automatically determine the weight of each
component in product spaces. Extensive experiments on synthetic and real-world
graph datasets demonstrate that WEIGHTED-PM is capable of learning better graph
representations with lower geometric distortion from input data, and performs
better on multiple downstream tasks, such as word similarity learning, top-$k$
recommendation, and knowledge graph embedding.
