On distributional graph signals

TL;DR

This paper introduces a novel framework for graph signal processing that uses Wasserstein space to model uncertainty in signals, providing a more general and robust approach than traditional vector-based methods.

Contribution

It proposes the use of Wasserstein space for distributional graph signals, unifying existing theories of graph uncertainty and developing new processing tools.

Findings

Wasserstein space generalizes classical graph signal space

Framework effectively models signal stochasticity

Demonstrated applicability on real datasets

Abstract

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.