Graph Signal Representation with Wasserstein Barycenters

TL;DR

This paper introduces a novel non-linear graph signal representation method using Wasserstein barycenters, leveraging optimal transport theory to improve low-dimensional analysis of signals on weighted graphs.

Contribution

It proposes a new geometry-aware, non-linear representation framework for graph signals based on Wasserstein barycenters, moving beyond traditional linear dictionary approaches.

Findings

Effective low-dimensional representations demonstrated

Outperforms linear methods in capturing graph signal structure

Potential for improved data analysis on graphs

Abstract

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.