Optimal Wiener-Filter Solutions for Denoising of Graph Signals on Directed Graphs

Graph signal processing has opened new avenues to the canonical denoising problem in interesting settings. Specifically, here we propose a Wiener-filter solution for graph signals on directed graphs. Under various stationarity assumptions combining uncorrelated and correlated noise conditions, we show optimal solutions, including a successful proof-of-concept for temperature graph.

Statistical Testing on Directed Graphs by Surrogate Data Generation

In recent years, graph signal processing has emerged as a powerful framework at the intersection of signal processing and graph theory, providing tools for the analysis of signals defined on nodes while accounting for their relationships represented by edges. These tools have been successfully applied to various settings, including statistical hypothesis testing. In particular, non-parametric approaches based on surrogate generation have been proposed for signals on undirected graphs. However, they are yet to be extended to directed graphs. In this work, we first revisit the notion of stationary graph signals on directed graphs. Specifically, and through the eigendecomposition of the graph shift operator, we define directed graph wide-sense stationary signals. Then, we propose a new framework to generate surrogate graph signals that preserve covariance structure under stationarity assumptions. Null distributions of the test metric can then be constructed from these surrogates and serve as a reference for the empirical data. Finally, we provide guiding examples and an application on real data, in which we compare the performance of our framework with existing techniques for undirected graphs or based on naive permutation, demonstrating feasibility and superiority of the proposed approach.

Hilbert Transform on Graphs: Let There Be Phase

In the past years, many signal processing operations have been successfully adapted to the graph setting. One elegant and effective approach is to exploit the eigendecomposition of a graph shift operator (GSO), such as the adjacency or Laplacian operator, to define a graph Fourier transform when projecting graph signals on the corresponding basis. However, the extension of this scheme to directed graphs is challenging since the associated GSO is non-symmetric and, in general, not diagonalizable. Here, we build upon a recent framework that adds a minimal number of edges to allow diagonalization of the GSO and thus provide a proper graph Fourier transform. We then propose a generalization of the Hilbert transform that leads to a number of simple and elegant recipes to effectively exploit the phase information of graph signals provided by the graph Fourier transform. The feasibility of the approach is demonstrated on several examples.