[Paper Review] Graph Fourier Transform based on Directed Laplacian

Rahul Singh, Abhishek Chakraborty+1 more|arXiv (Cornell University)|Jan 13, 2016

Advanced Graph Neural Networks24 references5 citations

TL;DR

This paper redefines the Graph Fourier Transform (GFT) within the Discrete Signal Processing on Graphs (DSP G) framework by using the Jordan eigenvectors and eigenvalues of the directed Laplacian matrix as graph harmonics and frequencies, respectively. It introduces a shift operator based on the directed Laplacian, enabling natural frequency ordering via total variation and ensuring LSI filters are polynomial in the directed Laplacian, thus unifying DSP G with traditional Laplacian-based GFT while improving interpretability.

ABSTRACT

In this paper, we redefine the Graph Fourier Transform (GFT) under the DSP$_\mathrm{G}$ framework. We consider the Jordan eigenvectors of the directed Laplacian as graph harmonics and the corresponding eigenvalues as the graph frequencies. For this purpose, we propose a shift operator based on the directed Laplacian of a graph. Based on our shift operator, we then define total variation of graph signals, which is used in frequency ordering. We achieve natural frequency ordering and interpretation via the proposed definition of GFT. Moreover, we show that our proposed shift operator makes the LSI filters under DSP$_\mathrm{G}$ to become polynomial in the directed Laplacian.

Motivation & Objective

• To address the lack of intuitive frequency interpretation in existing DSP G frameworks for directed graphs with complex or negative weights.
• To establish a natural frequency ordering in graph signal processing by redefining the Graph Fourier Transform (GFT) using the directed Laplacian matrix.
• To bridge the gap between the DSP G framework and the classical Laplacian-based GFT by aligning frequency interpretation with physical intuition.
• To ensure that Linear Shift-Invariant (LSI) filters in the DSP G framework are polynomial in the directed Laplacian, preserving mathematical consistency.

Proposed method

• Proposes a shift operator derived from the directed Laplacian matrix, replacing the standard weight matrix used in DSP G.
• Defines total variation of a graph signal based on the proposed shift operator to enable frequency ordering.
• Uses Jordan eigenvectors of the directed Laplacian as the graph Fourier basis and corresponding eigenvalues as graph frequencies.
• Reconstructs the GFT under the DSP G framework using the new shift operator and frequency ordering mechanism.
• Demonstrates that LSI graph filters become polynomials in the directed Laplacian under the proposed framework.
• Validates the natural frequency interpretation by showing that constant signals have only a zero-frequency component in the spectral domain.

Experimental results

Research questions

• RQ1 Can a natural frequency ordering be achieved in the DSP G framework for directed graphs with complex or negative edge weights?
• RQ2 How can the shift operator in DSP G be redefined to ensure intuitive frequency interpretation and link to the Laplacian-based GFT?
• RQ3 Does redefining GFT using the directed Laplacian ensure that LSI filters remain polynomial in the Laplacian matrix?
• RQ4 Is the zero-frequency component correctly identified as corresponding to constant graph signals under the new GFT definition?

Key findings

• The proposed GFT using the directed Laplacian's Jordan eigenvectors as graph harmonics results in a natural frequency ordering where small eigenvalues correspond to low frequencies.
• The total variation defined via the new shift operator enables consistent and intuitive frequency ordering, with equal variation for complex conjugate eigenvalues.
• The eigenvalue λ = 7.646 corresponds to the highest frequency in the graph, as confirmed by the magnitude spectrum of a test signal.
• A constant graph signal f = [1 1 1 1 1]^T yields only a single non-zero coefficient at zero frequency, confirming the zero-frequency component is correctly identified.
• LSI filters in the new framework are polynomial in the directed Laplacian, satisfying the necessary commutativity condition for shift-invariance.
• The Jordan eigenvector corresponding to the zero eigenvalue is the uniform vector 1/√N [1 1 ... 1]^T, confirming the zero-frequency interpretation.