  In many applications signals reside on the vertices of weighted graphs. Thus,
there is the need to learn low dimensional representations for graph signals
that will allow for data analysis and interpretation. Existing unsupervised
dimensionality reduction methods for graph signals have focused on dictionary
learning. In these works the graph is taken into consideration by imposing a
structure or a parametrization on the dictionary and the signals are
represented as linear combinations of the atoms in the dictionary. However, the
assumption that graph signals can be represented using linear combinations of
atoms is not always appropriate. In this paper we propose a novel
representation framework based on non-linear and geometry-aware combinations of
graph signals by leveraging the mathematical theory of Optimal Transport. We
represent graph signals as Wasserstein barycenters and demonstrate through our
experiments the potential of our proposed framework for low-dimensional graph
signal representation.
