Graph Fourier Transform (GFT) Overview

• Graph Fourier Transform (GFT) is a framework that extends Fourier analysis to graph signals by leveraging eigendecomposition of graph Laplacians and related matrices.
• It employs techniques like Jordan decomposition, SVD, and polar methods to handle both undirected and directed graph structures.
• GFT underpins practical applications in filtering, denoising, and recovery while enabling advanced analysis in graph signal processing and GNNs.

The Graph Fourier Transform (GFT) generalizes the classical discrete Fourier transform (DFT) to signals defined on the nodes of a graph, capturing frequency analysis, filtering, and spectral decomposition in networks with complex topology. In the GFT framework, the underlying graph structure replaces traditional time or spatial regularity, leveraging the eigendecomposition of graph operators (typically Laplacians or adjacency matrices) to define spectra, Fourier modes, and graph frequencies. GFT constructions have been extensively developed for undirected graphs and, more recently, for directed graphs, often relying on various matrix representations, optimization-theoretic formulations, and spectral characterizations to address the intricacies of network directionality, asymmetry, and singularity.

1. Fundamental Principles and Classical GFT

The foundational classical GFT is defined for undirected, weighted graphs $\mathcal{G} = (\mathcal{V}, \mathcal{E}, W)$ with $N = |\mathcal{V}|$ nodes, adjacency matrix $W$, and degree matrix $D = \mathrm{diag}(d_1, ..., d_N)$ where $d_i = \sum_j W_{ij}$. The combinatorial Laplacian is $L = D - W$, a symmetric positive semidefinite matrix admitting orthonormal eigendecomposition $L = U \Lambda U^T$, with $U = [u_1, ..., u_N]$ and $\Lambda = \mathrm{diag}(\lambda_1, ..., \lambda_N)$, $0 = \lambda_1 \leq ... \leq \lambda_N$. The GFT of a signal $N = |\mathcal{V}|$0 is $N = |\mathcal{V}|$1 and the inverse is $N = |\mathcal{V}|$2; the eigenvectors $N = |\mathcal{V}|$3 are the "graph Fourier modes,” and the eigenvalues $N = |\mathcal{V}|$4 are interpreted as "graph frequencies" governing the smoothness of the corresponding mode. Modes with smaller $N = |\mathcal{V}|$5 represent slowly varying (low-frequency) patterns, while larger $N = |\mathcal{V}|$6 reflect more oscillatory structure (Nitani et al., 17 Mar 2025).

For directed graphs, the Laplacian becomes asymmetric, complicating the spectral theory and necessitating alternative approaches for defining the GFT.

2. GFTs on Directed Graphs: Laplacian, SVD, Jordan, and Polar Approaches

Directed graphs introduce nonsymmetric adjacency and Laplacian matrices. Several seminal GFT construction strategies for digraphs include:

Directed Laplacian (Jordan-based) GFT

Given a digraph with adjacency $N = |\mathcal{V}|$7 (where $N = |\mathcal{V}|$8 reflects the edge $N = |\mathcal{V}|$9), in-degree matrix $W$0, the directed Laplacian is $W$1. As $W$2 is typically nondiagonalizable (asymmetric), the graph harmonics are chosen as the (generalized) Jordan eigenvectors $W$3 of $W$4 with associated eigenvalues $W$5, where $W$6 (Singh et al., 2016). The GFT is then defined as $W$7 for a signal $W$8, with the inverse $W$9, $D = \mathrm{diag}(d_1, ..., d_N)$0 being the Jordan basis of $D = \mathrm{diag}(d_1, ..., d_N)$1. Frequency ordering is obtained by considering the total variation $D = \mathrm{diag}(d_1, ..., d_N)$2; for normalized harmonics $D = \mathrm{diag}(d_1, ..., d_N)$3, $D = \mathrm{diag}(d_1, ..., d_N)$4, yielding a smoothness-to-oscillation spectrum.

SVD-based GFT

The SVD-based approach, applicable to general directed Laplacians or unified graph representation matrices (\textit{UGRM}s), decomposes the (potentially non-normal) matrix $D = \mathrm{diag}(d_1, ..., d_N)$5 (or $D = \mathrm{diag}(d_1, ..., d_N)$6 in the UGRM framework) as $D = \mathrm{diag}(d_1, ..., d_N)$7. Here, $D = \mathrm{diag}(d_1, ..., d_N)$8 contains nonnegative singular values ordered as graph frequencies, $D = \mathrm{diag}(d_1, ..., d_N)$9 (resp.\ $d_i = \sum_j W_{ij}$0) are right (left) singular vectors. The GFT is then defined by mixing these bases, e.g. $d_i = \sum_j W_{ij}$1, yielding a unitary transform with Parseval equality even when $d_i = \sum_j W_{ij}$2 is asymmetric (Chen et al., 2022, Xie et al., 12 Oct 2025).

Such SVD constructions generalize classical GFTs, with convergence to known DFTs for circulant digraphs, and facilitate efficient implementation and parameterized family definitions (UGRM-GFT) which interpolate between degree, Laplacian, and adjacency representations (Xie et al., 12 Oct 2025).

Polar Decomposition GFTs

Alternative digraph GFT designs leverage the polar decomposition of the adjacency matrix $d_i = \sum_j W_{ij}$3, where $d_i = \sum_j W_{ij}$4, $d_i = \sum_j W_{ij}$5 are unitary, $d_i = \sum_j W_{ij}$6, $d_i = \sum_j W_{ij}$7 are Hermitian positive semi-definite. Three GFTs are derived from the eigendecomposition of $d_i = \sum_j W_{ij}$8, $d_i = \sum_j W_{ij}$9, and $L = D - W$0, each associated with well-defined node-domain variation metrics—common-in-link, common-out-link, and shift-residual, respectively. This yields mutually orthogonal transforms, resolving non-orthogonality and numerical instability issues inherent in Jordan-based GFTs (Shimabukuro et al., 2023).

Projector-based and Variation-minimizing GFTs

For defective or non-diagonalizable adjacency matrices, the projector-based GFT decomposes signals into spectral projectors onto Jordan subspaces of $L = D - W$1, admitting generalized Parseval equality and unique, coordinate-free harmonic definitions (Deri et al., 2017). Lovász extension and related convex relaxations are exploited to define orthonormal GFT bases for digraphs by minimizing piecewise-linear (l1-type) directed variations, with tractable optimization (e.g., SOC, PAMAL) yielding numerically stable, cluster-constant transforms (Sardellitti et al., 2016).

3. GFTs for Product and Multi-dimensional Graphs

Large-scale graph signals often reside on cartesian product graphs (e.g., space × time), requiring multi-dimensional or joint GFTs. For graphs $L = D - W$2, $L = D - W$3 with Laplacians $L = D - W$4, $L = D - W$5, the joint Laplacian is $L = D - W$6, leading to Kronecker-factorized GFTs with spectral indices as frequency tuples $L = D - W$7 (Kurokawa et al., 2017, Cheng et al., 2022). SVD-based methods allow either full product SVD (global basis, high expressiveness, high cost) or separable Kronecker-GFT (lower complexity, effective for locally coupled signals) (Xie et al., 12 Oct 2025). These frameworks support multidimensional filtering, stationarity, and invariant analysis, and enable efficient denoising and subspace projections for spatiotemporal data (Cheng et al., 2022).

4. Graph Frequencies, Frames, and Basis Design

One challenge in GFT analysis is the nonuniform distribution and possible localization of GFT frequencies/eigenvalues—contrasting with the uniform frequency grid of the DFT. To address biases in spectral sampling and sparse representation, overcomplete frames (denser-graph-frequency GFFs/DGFFs) have been proposed. These augment the usual GFT basis with additional oscillating vectors interpolating intermediate frequencies, constructed via linear or analytic interpolation, yielding improved dispersion, sparser representations, and boosting filtering performance (Nitani et al., 17 Mar 2025).

Spread-frequency GFTs for directed graphs aim to construct orthonormal bases with directed variation (DV) frequencies as evenly spread as possible over the attainable frequency range. This is achieved via nonconvex dispersion minimization on the Stiefel manifold, or scalable greedy algorithms exploiting supermodularity and matroid constraints, with provable approximation guarantees (Shafipour et al., 2017, Shafipour et al., 2018).

5. Computational and Numerical Considerations

The dense eigendecomposition required by standard GFTs incurs $L = D - W$8 time and memory complexity ($L = D - W$9: number of nodes). Fast GFTs use symmetry, bipartition, and Haar/butterfly structures to reduce costs to $L = U \Lambda U^T$0 on graphs with suitable structure (e.g., symmetric lines, cycles, grids) (Lu et al., 2019). On arbitrary graphs, truncated Jacobi algorithms provide approximate fast GFTs controlled by the number of Givens rotations, enabling trade-offs between accuracy, storage, and run-time (LeMagoarou et al., 2017). Numerical stability on digraphs is tackled via optimization-based frameworks (e.g., the Stable Graph Fourier Algorithm, SGFA), which guarantee well-conditioned spectral bases through alternating projections and contraction mapping, systematically balancing diagonalization error and minimum singular value (Domingos et al., 2020).

6. GFT in Sampling, Recovery, and Signal Processing Applications

The GFT underpins sampling and recovery theory for graph signals, especially in settings where signals are bandlimited in the graph spectrum. In domains like water distribution networks, where dynamics exhibit low-rank structure rather than strict Laplacian bandlimiting, data-driven GFTs construct bases tailored to observed rank, guaranteeing perfect recovery from a minimal set of node measurements—a threshold often unattainable by classical or compressed sensing-based approaches (Wei et al., 2019). Denser spectral frames and overcomplete GFFs further enhance reconstruction, denoising, and inpainting, while facilitating signal-adaptive filter design (Nitani et al., 17 Mar 2025).

GFTs directly inform the design and spectral interpretation of graph filters—LSI filters are implemented as polynomials in the graph shift operator (Laplacian, adjacency, UGRM), with the spectral response specified by polynomial evaluation on (generalized) graph frequencies. This framework extends naturally to convolutional architectures in GNNs, enabling spectral kernel learning aligned to graph topology (Singh et al., 2016, Xie et al., 12 Oct 2025).

7. Extensions and Theoretical Generalizations

Recent work connects finite-graph GFTs to continuum analogues—the graphon Fourier transform—by analyzing the convergence of GFT spectra and modes as graph size grows, illuminating the transferability of spectral analysis and filter design to infinite dense-graph families (Ruiz et al., 2019). Generalized GFTs decouple the definitions of signal variation and energy, introducing flexible, application-driven orthonormal bases via generalized eigenproblems parameterized by positive-definite matrices reflecting sampling irregularity, energy metrics, or domain-specific structure (Girault et al., 2018). For localized or task-aware analysis, sparse GFTs optimize orthogonality, energy capture, and vertex sparsity via lasso-regularized regression reformulations (Safavi et al., 2018), providing interpretable, subgraph-focused spectral components.

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In summary, the Graph Fourier Transform provides a unifying spectral framework for signal processing on graphs, with diverse formulations and extensions handling directed and undirected graphs, multidimensional domains, irregular topologies, and domain-specific requirements. Technical advances in basis design, spectral frame enrichment, fast algorithms, and robust optimization underpin practical applications in sampling, recovery, denoising, and GNNs, while also establishing connections to broader spectral and functional analytic theory. Key milestones include Laplacian and SVD-based GFTs for digraphs (Singh et al., 2016, Chen et al., 2022, Xie et al., 12 Oct 2025), spread-frequency frameworks (Shafipour et al., 2017, Shafipour et al., 2018), numerically stable spectral optimization (Domingos et al., 2020), and denser-frequency frames (Nitani et al., 17 Mar 2025), consolidating the GFT’s foundational role in modern graph signal processing.