MGFRFT: Multi-dimensional Graph FRFT

• Multi-dimensional Graph Fractional Fourier Transform (MGFRFT) is a generalized transform for graph signals that performs independent fractional spectral analysis along each axis.
• It supports both undirected and directed graphs via eigendecomposition or SVD, facilitating closed-form and learnable filtering schemes for denoising, compression, and restoration.
• Its flexible parameterization and computational efficiency enable practical multi-dimensional spectral manipulation in high-dimensional, heterogeneous signal applications.

The Multi-dimensional Graph Fractional Fourier Transform (MGFRFT) generalizes the classical fractional Fourier transform (FRFT) to signals defined on Cartesian products of graphs, allowing independent fractional spectral analysis along each axis or domain of a multi-dimensional graph signal. This framework addresses the limitations of one-dimensional or single-parameter GFRFTs, enabling flexible adaptation to heterogeneous signal behaviors. MGFRFT encompasses both undirected and directed graphs via eigendecomposition or singular value decomposition (SVD), supports arbitrary order parameterization per dimension, and admits both closed-form and learnable filtering schemes. Notable specializations include the two-dimensional Graph Bi-Fractional Fourier Transform (2D-GBFRFT), multi-parameter GFRFTs, and geodesic-coupled variants that interpolate between standard time and graph-based fractional bases. MGFRFT has demonstrated superior denoising, compression, and restoration performance on synthetic, spatiotemporal, and real-world datasets.

1. Mathematical Definition and Transform Structure

Let $G_n = (\mathcal{V}_n, \mathbf{A}_n)$ for $n=1,\dots,M$ denote $M$ simple, undirected weighted graphs with adjacency matrices $\mathbf{A}_n$, each admitting the eigendecomposition

$$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$

where $$\boldsymbol{\chi}_n \in \mathbb{R}^{N_n \times N_n}$$ is orthonormal and $$\boldsymbol{\Lambda}_n = \mathrm{diag}(\lambda^{(n)}_1, \dots, \lambda^{(n)}_{N_n})$$. The fractional-order transform on $G_n$ with continuous order $\alpha_n \in \mathbb{R}$ is

$$\mathbf{F}_{G_n}^{\alpha_n} = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n^{\alpha_n} \boldsymbol{\chi}_n^T, \quad \boldsymbol{\Lambda}_n^{\alpha_n} = \mathrm{diag}((\lambda_1^{(n)})^{\alpha_n}, \dots, (\lambda_{N_n}^{(n)})^{\alpha_n})$$

For a signal $n=1,\dots,M$0 defined on the vertex set of a Cartesian product graph, the MGFRFT applies these transforms along each axis and stacks them via Kronecker products: $n=1,\dots,M$1 In the 2D-GBFRFT case ($n=1,\dots,M$2), for $n=1,\dots,M$3, $n=1,\dots,M$4: $n=1,\dots,M$5 For an $n=1,\dots,M$6, the transform is

$n=1,\dots,M$7

The framework accommodates both Laplacian- and adjacency-based constructions (Wang et al., 13 Oct 2025, Yan et al., 2021, Wang et al., 2 Mar 2026).

For directed graphs, the Laplacian $n=1,\dots,M$8 is decomposed via SVD: $n=1,\dots,M$9 and multi-dimensional transforms can use Kronecker products or Kronecker sums of the left/right singular vectors (Li et al., 4 Jun 2025).

2. Theoretical Properties

Key properties of MGFRFT and 2D-GBFRFT:

• Identity: $M$0
• Reduction: For $M$1, recovers single-order 2D-GFRFT: $M$2
• Invertibility and Index Additivity:

$M$3

and more generally,

$M$4

• Unitarity / Parseval: If graph eigenvectors are orthonormal (e.g., symmetric adjacency/Laplacian), the MGFRFT is unitary:

$M$5

• Linearity: The transform is linear in the input.
• Bandlimitedness: MGFRFT provides a framework for defining multi-dimensional or multi-parameter bandlimitedness.

These properties guarantee well-posedness of inversion, stable energy representation, and support for efficient spectral manipulation (Wang et al., 13 Oct 2025, Yan et al., 2021, Li et al., 4 Jun 2025, Cui et al., 31 Jul 2025).

3. Filtering, Parameter Estimation, and Learning

MGFRFT domains admit specialized filtering strategies that exploit their multi-dimensional separability and parametrization:

• Wiener-style Filtering (Grid Search): For denoising with observation model $M$6, filtering proceeds by

$M$7

and minimizing expected MSE over a grid of ($M$8), optimizing $M$9 via a Wiener system (Wang et al., 13 Oct 2025).

• Differentiable Joint Optimization: Orders ($\mathbf{A}_n$0) and the diagonal filter vector ($\mathbf{A}_n$1) can be optimized jointly by gradient descent:

$\mathbf{A}_n$2

exploiting differentiability with respect to fractional orders (Wang et al., 13 Oct 2025).

• Learnable Order Vectors: Multi-parameter GFRFTs (MPGFRFT) generalize this by learning a vector of orders over spectral components, with closed-form gradients supporting end-to-end adaptation for compression or denoising tasks (Cui et al., 31 Jul 2025).
• Hybrid Time-Vertex/Graph Interpolation: Hybridization with the joint time-vertex FRFT (JFRFT) leverages a convex combination parameter ($\mathbf{A}_n$3) to interpolate between classical time-based FRFT and graph-based FRFT bases.

These frameworks enable adaptation to both structural (graph) and temporal variability in complex signals (Wang et al., 13 Oct 2025, Cui et al., 31 Jul 2025, Wang et al., 2 Mar 2026).

4. Hybrid and Coupled Transform Extensions

A notable generalization is the geodesic-coupled GFRFT (GC-GFRFT), which interpolates between two unitary bases (e.g., graph-induced and classical FRFT) along the principal geodesic of the unitary group. For spatial order $\mathbf{A}_n$4, temporal order $\mathbf{A}_n$5, and coupling $\mathbf{A}_n$6, the transform is

$\mathbf{A}_n$7

with

$\mathbf{A}_n$8

yielding a unitary transform that at $\mathbf{A}_n$9 is the 2D-GBFRFT and at $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$0 is the JFRFT (Wang et al., 2 Mar 2026, Wang et al., 13 Oct 2025).

This coupling enables unified control over spectral bases, allowing the transform to adaptively select the optimal blend of graph and classical temporal analysis.

5. Computational Complexity and Implementation

MGFRFTs offer significant computational advantages when acting on product graphs:

• Eigendecomposition: $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$1 for the factors, versus $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$2 for a full product graph.
• Forward/Inverse Transform: Applied sequentially or via tensor contractions, yields $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$3 total complexity for $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$4 factors and $$\mathbf{A}_n = \boldsymbol{\chi}_n \boldsymbol{\Lambda}_n \boldsymbol{\chi}_n^T$$5 (Yan et al., 2021, Li et al., 4 Jun 2025).
• Fast Implementation: For regular graphs (e.g., paths, grids), fast transforms analogous to the FFT can be used. For large sparse graphs, Lanczos or partial EVD is effective.
• Storage: No need to instantiate large Kronecker matrices explicitly; transforms can be applied dimension-wise via mode-multiplication (Li et al., 4 Jun 2025).

These efficiencies make MGFRFT practical for high-dimensional or large-scale graph signals.

6. Practical Applications and Experimental Findings

MGFRFT, and in particular 2D-GBFRFT and its variants, have been evaluated in a range of graph signal processing tasks:

• Compression: On spatiotemporal weather data, MGFRFT yields highly sparse representations, with only a few percent of coefficients sufficient for low relative error and high PSNR (Yan et al., 2021).
• Denoising: Bandlimiting in the MGFRFT domain effectively denoises temperature and air-quality datasets, outperforming classical GFRFT and single-parameter methods; energy concentration is substantially higher for multi-dimensional transforms (Li et al., 4 Jun 2025).
• Image Restoration and Deblurring: For dynamic image deblurring (e.g., REDS’A/B datasets), 2D-GBFRFT produces lower MSE, higher PSNR, and sharper visual detail relative to 2D-GFRFT, especially around edges; hybrid transforms provide further improvement (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).
• Spectral Adaptivity: Hybrid time-vertex/graph transforms and geodesic coupling allow selection of the spectral basis most suited to the temporal or spatial inhomogeneity of the data, with the coupling parameter tuned structurally (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).
• Compression and Encryption: Multi-parameter MGFRFTs (Type I and II) show greatly improved performance for ultra-low-ratio compression and show key-sensitivity advantages for image encryption (Cui et al., 31 Jul 2025).

Consistently, MGFRFT-based approaches surpass one-order or non-fractional methods in representational efficiency, denoising, and restoration tasks across synthetic and real datasets.

7. Extensions and Relation to Other Transforms

MGFRFT frameworks encompass and generalize numerous previously proposed transforms:

• Single-order GFRFT: MGFRFT reduces to the standard GFRFT when all orders are identical.
• Joint Time-Vertex FRFT (JFRFT): A special case of the hybrid/interpolated scheme, acting on time–graph signals with coupled fractional orders (Wang et al., 13 Oct 2025).
• Multi-parameter GFRFTs (MPGFRFT-I/II): Permit non-constant order-vectors, further increasing adaptability at the expense of additional storage or computation, and supporting gradient-based learning schemes (Cui et al., 31 Jul 2025).
• SVD-based and Kronecker-sum/product MGFRFTs: For directed graphs, transforms constructed via SVD of fractional Laplacians extend the framework to non-symmetric structures (Li et al., 4 Jun 2025).

These connections position MGFRFT as a unifying scheme for multi-dimensional and flexible spectral analysis of graph-supported signals. The framework’s adaptability and efficiency have solidified its foundational role in modern graph signal processing (Wang et al., 13 Oct 2025, Yan et al., 2021, Wang et al., 2 Mar 2026, Li et al., 4 Jun 2025, Cui et al., 31 Jul 2025).