On the Graph Fourier Transform for Directed Graphs

TL;DR

This paper introduces a novel approach for defining the Graph Fourier Transform on directed graphs by minimizing a continuous extension of the graph cut size, with iterative algorithms to handle non-convexity and orthogonality constraints.

Contribution

It proposes a new method for constructing the GFT basis for directed graphs using Lovász extension and develops iterative algorithms to optimize this basis.

Findings

Effective algorithms for GFT basis construction on directed graphs.

Extension to balanced cut size minimization.

Handles non-convex optimization challenges efficiently.

Abstract

The analysis of signals defined over a graph is relevant in many applications, such as social and economic networks, big data or biological networks, and so on. A key tool for analyzing these signals is the so called Graph Fourier Transform (GFT). Alternative definitions of GFT have been suggested in the literature, based on the eigen-decomposition of either the graph Laplacian or adjacency matrix. In this paper, we address the general case of directed graphs and we propose an alternative approach that builds the graph Fourier basis as the set of orthonormal vectors that minimize a continuous extension of the graph cut size, known as the Lov\'{a}sz extension. To cope with the non-convexity of the problem, we propose two alternative iterative optimization methods, properly devised for handling orthogonality constraints. Finally, we extend the method to minimize a continuous relaxation of the balanced cut size. The formulated problem is again non-convex and we propose an efficient solution method based on an explicit-implicit gradient algorithm.