  Graph signal processing (GSP) studies graph-structured data, where the
central concept is the vector space of graph signals. To study a vector space,
we have many useful tools up our sleeves. However, uncertainty is omnipresent
in practice, and using a vector to model a real signal can be erroneous in some
situations. In this paper, we want to use the Wasserstein space as a
replacement for the vector space of graph signals, to account for signal
stochasticity. The Wasserstein is strictly more general in which the classical
graph signal space embeds isometrically. An element in the Wasserstein space is
called a distributional graph signal. On the other hand, signal processing for
a probability space of graphs has been proposed in the literature. In this
work, we propose a unified framework that also encompasses existing theories
regarding graph uncertainty. We develop signal processing tools to study the
new notion of distributional graph signals. We also demonstrate how the theory
can be applied by using real datasets.

Signal Processing