Beyond Single-Window Graph Fourier Analysis

TL;DR

Addresses joint vertex–frequency analysis of signals on graphs by introducing MWGFT, a multi-window extension of the WGFT based on generalized translation and modulation induced by the graph Laplacian. It derives exact reconstruction formulas for complex-valued signals with multiple analysis and synthesis windows and proves frame properties guaranteeing stable inversion. Numerical experiments on synthetic and real graphs confirm exact reconstruction up to machine precision and show improved vertex–frequency localization compared with single-window methods, especially on irregular topologies. The approach offers a flexible, frame-theoretic tool for localized spectral analysis on networks and points to scalable extensions using approximate spectral filtering for large graphs.

Abstract

We introduce a multi-windowed graph Fourier transform (MWGFT) for the joint vertex-frequency analysis of signals defined on graphs. Building on generalized translation and modulation induced by the graph Laplacian, the proposed framework extends the windowed graph Fourier transform by allowing multiple analysis and synthesis windows. Exact reconstruction formulas are derived for complex-valued graph signals, together with sufficient and computable conditions guaranteeing stable invertibility. The associated families of windowed graph Fourier atoms are shown to form frames for the space of graph signals. Numerical experiments on synthetic and real world graphs confirm exact reconstruction up to machine precision and demonstrate improved stability and vertex-frequency localization compared to single-window constructions, particularly on irregular graph topologies.

Figures (8)

• Figure 1: Original and reconstructed signals on the graph. The left panel shows the original signal $f$ defined on the graph vertices, while the right panel displays the reconstructed signal $f_{rec}$ obtained via the multi-windowed graph Fourier transform synthesis formula. The close visual agreement confirms exact signal recovery up to numerical precision in the vertex domain.
• Figure 2: Spectral analysis windows $\widehat{g_j}(\lambda)$ (left) and corresponding synthesis windows $\widehat{\gamma_j}(\lambda)$ (right) in the graph Fourier domain.
• Figure 3: MWGFT spectrograms of an impulse on a path graph. From left to right: averaged spectrogram over the three windows, and spectrograms corresponding to each individual window. Here, the vertical axis corresponds to graph vertices, while the horizontal axis indexes graph Fourier modes (i.e., Laplacian eigenmodes).
• Figure 4: Original and reconstructed signals on the graph. The left panel shows the original signal $f$ defined on the graph vertices, while the right panel displays the reconstructed signal $f_{rec}$ obtained via the multi-windowed graph Fourier transform synthesis formula. The close visual agreement confirms exact signal recovery in the vertex domain. Third row shows pointwise reconstruction error $|f-f_{rec}|$. The error remains at numerical precision, confirming the stability of the reconstruction.
• Figure 5: Spectral windows and corresponding reconstruction denominators. Top row: spectral profiles of the analysis windows $\widehat{g}_j(\lambda)$ (RBF kernel type) for window indices $j=1,3,5$. Bottom row: corresponding values of the reconstruction denominator $\left|\sum_{j} \langle T_n \gamma_j,\, T_n g_j\rangle\right|$ as a function of the vertex index $n$. The denominator remains strictly positive for all vertices, ensuring stable reconstruction.

Theorems & Definitions (27)

• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5