Graph Fourier Transform Overview

• Graph Fourier Transforms are mathematical tools that generalize classical Fourier analysis to graph signals by using Laplacian eigen-decomposition and tailored basis for various graph structures.
• They enable frequency-domain analysis on irregular domains, supporting applications such as image coding, spectral clustering, and graph neural networks through directional and energy-based filtering.
• Efficient computation is achieved via sparse factorizations, SVD-based methods, and approximate diagonalization techniques that balance numerical stability with reduced computational complexity.

A graph Fourier transform (GFT) generalizes the classical Fourier transform to functions defined on the vertices of a graph, providing a frequency-domain analysis tool for signals on arbitrary graph topologies—including undirected, directed, weighted, and product graphs. GFTs are fundamental to graph signal processing, enabling definitions of frequency, convolution, filtering, and sampling in irregular domains. Unlike the classical setting, GFT construction is sensitive to graph symmetry, directionality, and irregularity, resulting in a rich taxonomy of transform definitions tailored to different graph structures and signal variation models.

1. Mathematical Formulation and Frequency Ordering

For an undirected weighted graph $G=(V, E, W)$, the combinatorial Laplacian $L = D - W$ is real symmetric and positive semidefinite. It has an orthonormal eigendecomposition $L = U \Lambda U^\top$, yielding GFT basis vectors as the eigenvectors $\{u_k\}$ and frequencies as eigenvalues $\{\lambda_k\}$ (Girault et al., 2019). The graph Fourier transform of $x$ is $\hat{x} = U^\top x$, and the inverse is $x = U \hat{x}$. Frequency ordering is naturally induced by the Laplacian quadratic form, with low (small) $\lambda_k$ corresponding to smooth modes and high $\lambda_k$ to oscillatory ones.

For directed graphs, conventional Laplacian-based approaches fail due to non-symmetry and non-diagonalizability. Several methods exist:

• Jordan Decomposition: For a directed Laplacian $L = D_{\text{in}} - W$, the GFT uses a Jordan basis $V$ for $L = V J V^{-1}$ (Singh et al., 2016). The transform is $\hat x = V^{-1} x$, inverse $x = V \hat x$. Frequencies are the eigenvalues $\lambda_k$, possibly complex, and frequency ordering is defined via the $L$-induced total variation: $TV_G(v_k) \propto |\lambda_k|$ upon $\ell_1$ normalization.
• SVD-based GFTs: Given possibly non-diagonalizable $L$, use the thin SVD $L = U\Sigma V^\top$ where $\Sigma$ holds non-negative singular values $\{\sigma_k\}$. The SVD-based GFT maps $x \mapsto \frac{1}{2}(U^\top + V^\top)x$ and $(U^\top - V^\top)x$ (Chen et al., 2022, Cheng et al., 2022). This approach is numerically stable and reduces to the classical eigendecomposition in the symmetric case.
• Spectral Projector/Generalized Eigenspace: The GFT can be formulated via spectral projectors onto Jordan subspaces (Deri et al., 2017, Deri et al., 2017, Deri et al., 2017). Here, the transform extracts projections onto each minimal $A$-invariant subspace, leading to a coordinate-free decomposition and well-defined total-variation-based frequency ordering, even when the adjacency matrix is defective.
• Polar Decomposition: For directed graphs, the SVD $A=U\Sigma V^\top$ enables the polar factorization $A = P Q = Q F$, from which three GFTs can be defined via $P$, $F$, and $Q$ decompositions, corresponding to "common-in-link", "common-out-link", and "in-flow" modes of variation, respectively (Shimabukuro et al., 2023).

2. GFT for Structured and Product Graphs

For graphs exhibiting symmetries or product structure, the GFT admits further specialization and computational accelerations:

• Symmetric Grids/Image Blocks: Nodes arranged on regular grids can be modeled as graphs with designed edge symmetries, enabling symmetry-based GFTs (SBGFTs) that produce directional, non-separable bases and exploit block-diagonalization for fast implementations (Gnutti et al., 24 Nov 2024). The Laplacian becomes centrosymmetric, allowing eigenvector computation at half the cost of brute-force approaches.
• Cartesian Product Graphs: For $G = G_1 \square G_2$, the Laplacian $L_G = L_1 \otimes I + I \otimes L_2$ admits eigenvectors as Kronecker products of the factors' eigenvectors (Kurokawa et al., 2017). The multi-dimensional GFT (MGFT) arranges the spectrum in a tensor indexed by $(\lambda_{k_1}^{(1)}, \lambda_{k_2}^{(2)})$, enabling explicit directional frequency analysis. In the case of directed product graphs, SVD-based constructions provide two non-redundant GFT definitions, one via direct SVD of the product Laplacian and one via Kronecker products of the factors' SVDs (Cheng et al., 2022).
• Enveloping Cayley Digraphs: For arbitrary digraphs lacking a well-posed Fourier basis, envelope extensions embed the given digraph in a diagonalizable Cayley (circulant) digraph, yielding a DFT-like GFT that is numerically robust and supports convolution/algebraic shift-invariance. Optimal envelopes are chosen based on spectral/fidelity and numerical conditioning metrics (Bardi et al., 29 Jul 2024).

3. Efficient and Approximate Algorithms

The high computational cost of dense eigendecomposition motivates fast and approximate GFTs:

• Sparse Factorizations: For undirected graphs, fast GFT schemes factor the Fourier matrix $U$ into a product of sparse (Givens rotation or Haar) matrices. For graphs with bipartite or center symmetry, butterfly-stage decompositions achieve operation counts approaching $O(n \log n)$, with exactness or near-exactness on line, grid, or cycle graphs (Lu et al., 2019).
• Approximate Diagonalization: Greedy, Jacobi-type algorithms construct orthogonal approximations $\tilde U \approx U$ as products of a limited number $K$ of sparse Givens rotations. For $K = O(n \log n)$, one obtains transforms $\tilde U$ that diagonalize $L$ to high fidelity at $O(n \log n)$ vector-multiplication cost (Magoarou et al., 2016). Parallel blocking and truncation enable practical construction on large graphs.
• Iterative Low-complexity Eigenspace Construction: General matrices (including non-symmetric cases) can be approximately diagonalized with a fixed number of fundamental transformations (orthogonal or invertible on small subspaces). Jacobi-like refinement optimizes transform fidelity given a specified complexity budget, with exponential error decay in the number of factors (Rusu et al., 2020).
• Agile Inexact Methods (AIM): For large defective matrices, GFTs can be approximated by projections onto generalized eigenspaces rather than explicit Jordan chains, giving drastic reductions in execution time (orders of magnitude faster) with minimal fidelity loss for the dominant spectral content (Deri et al., 2017).

4. Graph Signal Variation and Energy

Signal variation and energy concepts are central to GFT construction and interpretation.

• The Laplacian quadratic form $$x^\top L x = \sum_{(i,j)\in E} w_{ij} (x_i - x_j)^2$$ quantifies signal smoothness on undirected graphs (Girault et al., 2019). Small eigenvalue eigenvectors are smooth; large eigenvalues indicate oscillatory content.
• On directed graphs, total variation is generalized to accommodate asymmetric structures. For Laplacian-based GFTs, $TV_G(x) = \|Lx\|_1$ is adopted, and on a normalized eigenvector $v_k$, this reduces to $|λ_k|$ after normalization (Singh et al., 2016).
• Generalizations allow independence in choosing the signal's inner product and variation operator, as in the irregularity-aware GFT framework (Girault et al., 2018). Here, any Hermitian PSD $M$ (variation) and positive-definite $Q$ (energy weighting) yield a spectrum via the generalized eigenproblem $M u_l = \lambda_l Q u_l$, enabling GFT adaptation to sampling irregularity, degree bias, or Voronoi-cell-weighted signal norms.
• For polar/SVD-based directed GFT, three node-domain variation metrics are defined: common-in-link, common-out-link, and in-flow, with frequency ordering reflecting smoothness relative to distinct node connectivity patterns (Shimabukuro et al., 2023).

5. Filtering, Convolution, and Invariance Properties

Graph filters $H$ are usually defined as polynomials in the shift operator (adjacency or Laplacian): $H = h(L) = \sum_{m=0}^{M-1} h_m L^m$ (Singh et al., 2016). Shift-invariance and diagonalizability of the graph operator (or its extension) guarantee that such filters admit spectral action: Fourier transforming $x$ yields $\hat{x}$, and filtering corresponds to entrywise (possibly blockwise) multiplication in the spectral domain.

For enveloped digraphs, convolution is defined spectrally: $x ⋆ y := V( (V^{-1} x) \circ (V^{-1} y) )$, with the convolution theorem $F(x ⋆ y) = Fx \circ Fy$ holding (Bardi et al., 29 Jul 2024). All polynomial filters in this algebra are convolution operators.

Eigenbasis or projector-based GFT constructions support generalized Parseval's identities, ensuring energy conservation across spectral components—even when eigenbases are not orthogonal (Deri et al., 2017, Deri et al., 2017).

6. Practical Applications and Implementations

GFTs are essential in graph-based denoising, compression, wavelet analysis, sampling, spectral clustering, and as inductive biases in graph neural architectures.

• Video and Image Coding: SBGFTs, with variable-size and axes-aligned symmetry adaptation, improve lower-bitrate image representation vs. DCT, yielding up to 9.3% BD-rate saving in VVC intra-coding with only marginal complexity increase (Gnutti et al., 24 Nov 2024). Fast stage-decomposed GFTs enable design of efficient, non-separable transforms suitable for pixel blocks.
• Spectral Clustering and Sampling: Irregularity-aware GFTs improve clustering performance under degree heterogeneity (Girault et al., 2018), and GFT visualization tools elucidate how localized spectral content and sampling irregularity interact (Girault et al., 2019).
• Graph Neural Networks: GFTs have entered attention mechanisms—Grafourierformer integrates node-wise frequency information and Laplacian eigenvalues to bias Transformer self-attention, improving discrimination of smooth (global) and oscillatory (local/noisy) graph patterns in node and graph classification benchmarks (Zhai et al., 28 Apr 2025).
• Sensor Networks and Spatio-Temporal Analysis: Multi-dimensional GFTs allow joint spatial-temporal denoising and filtering of data on product graphs, as in temperature field denoising on time×space networks, with significant computational savings over naive diagonalization (Kurokawa et al., 2017, Cheng et al., 2022).

Implementation typically requires efficient eigensolver access or fast approximate transforms. For undirected (sparse, structured) graphs, truncated Jacobi and Haar-stage factorization methods enable $O(n \log n)$ per-vector GFT with near-optimal error (Magoarou et al., 2016, Lu et al., 2019). For directed or defective graphs, robust SVD-based or generalized eigenspace methods are favored for numerical stability and scalability (Chen et al., 2022, Deri et al., 2017, Rusu et al., 2020).

7. Theoretical and Practical Considerations

• Basis and Label Invariance: Projector-based GFTs are coordinate-free and stable to node relabeling, with frequency ordering invariant under permutations and basis choices (Deri et al., 2017).
• Defective and Non-diagonalizable Operators: The choice of Jordan, SVD, or spectral projector framework directly impacts practical computability and numerical stability for graphs with repeated or defective eigenvalues.
• Numerical Stability: Condition number and basis approximation errors must be tightly controlled; envelope extension and polar/SVD-based methods are designed for this purpose (Bardi et al., 29 Jul 2024, Shimabukuro et al., 2023, Chen et al., 2022).
• Approximation vs. Complexity Tradeoff: For scalable GFT, aggressive factorization and inexact eigenspace grouping are deployed, striking explicit complexity/accuracy compromises (as in large urban networks or high-dimensional sensor arrays) (Rusu et al., 2020, Deri et al., 2017, Magoarou et al., 2016).
• Algorithmic Exploitation of Symmetry/Product Structure: Block-diagonalization and product Kronecker structures can be exploited for dramatic computational savings, often reducing GFT computation to eigenproblems on much smaller factor graphs (Kurokawa et al., 2017, Gnutti et al., 24 Nov 2024, Lu et al., 2019).
• Irregularity and Application-specific Optimization: Adaptation to graph irregularity (degree, sampling, Voronoi partition) is crucial in scientific and engineering contexts. The GFT can be parameterized (via the energy matrix $Q$ and the variation matrix $M$) to minimize bias and variance relative to the physical measurement model (Girault et al., 2018).

The field continues to advance along axes of faster computation, greater robustness to non-diagonalizability and irregularity, and integration with signal processing and learning tasks on highly complex graphs (Singh et al., 2016, Deri et al., 2017, Zhai et al., 28 Apr 2025).