Graph Signal Processing for Directed Graphs Based on the Hermitian Laplacian

Graph signal processing is a useful tool for representing, analyzing, and processing the signal lying on a graph, and has attracted attention in several fields including data mining and machine learning. A key to construct the graph signal processing is the graph Fourier transform, which is defined by using eigenvectors of the graph Laplacian of an undirected graph. The orthonormality of eigenvectors gives the graph Fourier transform algebraically desirable properties, and thus the graph signal processing for undirected graphs has been well developed. However, since eigenvectors of the graph Laplacian of a directed graph are generally not orthonormal, it is difficult to simply extend the graph signal processing to directed graphs. In this paper, we present a general framework for extending the graph signal processing to directed graphs. To this end, we introduce the Hermitian Laplacian which is a complex matrix obtained from an extension of the graph Laplacian. The Hermitian Laplacian is defined so as to preserve the edge directionality and Hermitian property and enables the graph signal processing to be straightforwardly extended to directed graphs. Furthermore, the Hermitian Laplacian guarantees some desirable properties, such as non-negative real eigenvalues and the unitarity of the Fourier transform. Finally, experimental results for representation learning and signal denoising of/on directed graphs show the effectiveness of our framework.