Angular Graph Fractional Fourier Transform

• AGFRFT is a spectral analysis framework that integrates fractional-order and angular-rotation techniques for processing signals on irregular graphs.
• It employs three controllable angles to modulate spatial and temporal spectral bases, ensuring unitary, invertible, and degenerate mappings to classical transforms.
• The method supports adaptive denoising and filtering on spatiotemporal graphs using grid search and gradient descent, achieving notable performance improvements.

The Angular Graph Fractional Fourier Transform (AGFRFT) is a spectral analysis tool synthesizing fractional-order and angular-rotation paradigms in graph signal processing. By parameterizing the spectral basis of a graph via continuous fractional and angular parameters, AGFRFT enables fine-grained, unitary, and invertible analysis of signals defined on irregular domains—including spatiotemporal product graphs—for denoising and adaptive spectral filtering. AGFRFT incorporates three jointly-controllable angles, which respectively modulate the fractional interpolation for spatial and temporal components, and interpolate unitarily between different families of temporal bases, all while ensuring theoretical consistency with degeneration to the classical graph Fourier transform (GFT).

1. Mathematical Foundations and Construction

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ denote an undirected graph with $N$ nodes and a graph-shift operator $Z \in \mathbb{C}^{N\times N}$ admitting an orthonormal eigendecomposition $Z = U\Lambda U^{H}$, $U^HU=I_N$. For a graph signal $x\in\mathbb{R}^N$, the GFT is $\hat{x} = F x$, with $F=U^H$.

The AGFRFT generalizes both the Graph Fractional Fourier Transform (GFRFT) and the Angular Graph Fourier Transform (AGFT):

• Fractional parameterization: For a fractional order $\alpha\in[0,1]$, define $F^\alpha := \exp(\alpha\log F)$; $N$0 recovers the identity, while $N$1 yields the standard GFT.
• Angular rotation: Construct a rotation matrix family $N$2 satisfying $N$3 and $N$4 smoothness in $N$5. This rotates the eigenbasis: $N$6, and $N$7.

Two AGFRFT variants are defined:

• Type I (I-AGFRFT): Uses $N$8
• Type II (II-AGFRFT): Rotates the fractional GFT basis; $N$9, $Z \in \mathbb{C}^{N\times N}$0

Degeneracy: Importantly, $Z \in \mathbb{C}^{N\times N}$1 and $Z \in \mathbb{C}^{N\times N}$2, ensuring that AGFRFT reduces to GFRFT or AGFT at the endpoints (Zhao et al., 20 Nov 2025).

For product graphs (e.g., spatiotemporal domains), the bi-fractional Fourier transform assigns independent fractional orders to the two factor graphs:

$Z \in \mathbb{C}^{N\times N}$3

Vectorized: $Z \in \mathbb{C}^{N\times N}$4 (Wang et al., 2 Mar 2026).

2. Geodesic Coupling and Three-Angle Parameterization

In spatiotemporal graphs, the temporal fractional basis admits two canonical endpoints:

• Graph-induced GFRFT ($Z \in \mathbb{C}^{N\times N}$5) on the temporal graph factor
• Classical Discrete Fractional Fourier Transform ($Z \in \mathbb{C}^{N\times N}$6), standard in signal processing

Rather than convexly combining these, AGFRFT employs the principal geodesic of the unitary group $Z \in \mathbb{C}^{N\times N}$7 to interpolate via a coupling parameter $Z \in \mathbb{C}^{N\times N}$8, providing a coupled temporal spectral basis:

$Z \in \mathbb{C}^{N\times N}$9

$Z = U\Lambda U^{H}$0 is always unitary for real $Z = U\Lambda U^{H}$1.

This yields a global geodesic-coupled GFRFT:

$Z = U\Lambda U^{H}$2

Setting $Z = U\Lambda U^{H}$3 recovers the pure bi-fractional transform; $Z = U\Lambda U^{H}$4 recovers the time-vertex DFRFT–GFRFT (Wang et al., 2 Mar 2026).

To unify the parameterization, AGFRFT uses

• $Z = U\Lambda U^{H}$5 as spatial and temporal fractional angles, $Z = U\Lambda U^{H}$6
• $Z = U\Lambda U^{H}$7 as the geodesic coupling "angle," $Z = U\Lambda U^{H}$8

The transform becomes:

$Z = U\Lambda U^{H}$9

3. Theoretical Properties

AGFRFT possesses several key properties:

• Unitarity: Both types (I, II) of AGFRFT are unitary: $U^HU=I_N$0.
• Invertibility: I-AGFRFT satisfies $U^HU=I_N$1. For II-AGFRFT, $U^HU=I_N$2.
• Degeneracy: Both types satisfy $U^HU=I_N$3 and recover GFRFT (for $U^HU=I_N$4) or AGFT (for $U^HU=I_N$5). All maps are $U^HU=I_N$6 in their parameters.
• Index Additivity: For Type I, $U^HU=I_N$7 (Zhao et al., 20 Nov 2025).

On product graphs, the joint transform remains unitary and invertible, guaranteeing preservation of energy and invertible spectral representations (Wang et al., 2 Mar 2026).

4. Learning and Optimization

AGFRFT supports end-to-end learnable joint parameterization of its angles, as well as the spectral filter, for adaptive graph signal processing:

• For small graphs, grid search over $U^HU=I_N$8 can identify optimal settings.
• For larger graphs, gradient descent is used; the entire mapping $U^HU=I_N$9 is differentiable by backpropagation through matrix logarithm and exponential.
• In a Wiener-type filtering context, one recasts signal estimation as

$x\in\mathbb{R}^N$0

and minimizes $x\in\mathbb{R}^N$1 via gradient descent, with updates for all parameters (Zhao et al., 20 Nov 2025). When AGFRFT is used in multi-factor (e.g., spatiotemporal) graphs, the fractional orders and the geodesic coupling angle can be learned from data end-to-end, while the coupling parameter may also be treated as a fixed regularizer (Wang et al., 2 Mar 2026).

5. Computational Methods

For practical computation:

• Grid Search: Suitable for small $x\in\mathbb{R}^N$2; discretize $x\in\mathbb{R}^N$3, construct $x\in\mathbb{R}^N$4, form $x\in\mathbb{R}^N$5, solve for the optimal Wiener-Hopf filter $x\in\mathbb{R}^N$6, and select the configuration yielding minimum MSE. Computational complexity is $x\in\mathbb{R}^N$7.
• Gradient-Based Methods: For larger $x\in\mathbb{R}^N$8, initialize $x\in\mathbb{R}^N$9 and use iterative gradient descent through the differentiable mappings. Complexity scales as $\hat{x} = F x$0 for $\hat{x} = F x$1 iterations, outperforming exhaustive search in efficient parameter discovery.

A plausible implication is that, due to full differentiability, AGFRFT-based architectures can be readily integrated into neural or end-to-end adaptive systems for graph signal learning (Zhao et al., 20 Nov 2025).

6. Empirical Evaluation and Applications

Extensive evaluations demonstrate AGFRFT’s capabilities:

• Time-Series Graph Denoising: On datasets such as SST and PM$\hat{x} = F x$2, II-AGFRFT achieves MSE reductions of $\hat{x} = F x$3 and $\hat{x} = F x$4 over GFRFT, with II-AGFRFT and I-AGFRFT outperforming AGFT in all cases.
• Image Denoising: On standard benchmarks (Set12, $\hat{x} = F x$5), II-AGFRFT achieves PSNR of $\hat{x} = F x$6 dB and MSE $\hat{x} = F x$7, surpassing GFRFT and AGFT by $\hat{x} = F x$8 dB and $\hat{x} = F x$9 dB in PSNR, respectively.
• Point-Cloud Denoising: II-AGFRFT improved PSNR by 5.13 dB and reduced MSE by $F=U^H$0 compared to GFRFT for Microsoft’s “David9” (patch size $F=U^H$1, $F=U^H$2) (Zhao et al., 20 Nov 2025).

By continuous variation of $F=U^H$3, AGFRFT interpolates between the spatial/vertex domain and graph-frequency domain. This enables spectral concentration improvements of $F=U^H$4 over either GFRFT or AGFT alone. Applications span spatiotemporal signal separation on networks, adaptive graph-convolutional neural layers, source localization, biomedical imaging on dynamic brain graphs, and patch-wise AGFRFT-based Wiener filtering for image restoration (Wang et al., 2 Mar 2026, Zhao et al., 20 Nov 2025).

7. Connections, Limitations, and Extensions

AGFRFT unifies and generalizes previous graph spectral transforms:

• AGFT introduces angular spectral control but previously failed to degenerate to the GFT at zero angle; GFRFT provides fractional flexibility but lacks angular modulation.
• AGFRFT, by employing a degeneracy-friendly rotation matrix family, achieves mathematically consistent merging of angular and fractional regimes.
• In the spatiotemporal setting, AGFRFT’s tri-angular (three-angle) scheme enables otherwise inaccessible interpolants between frequency-analysis bases, subsuming 2D-GFT, BGFRFT, and joint time-vertex FRFT as special instances (Wang et al., 2 Mar 2026).

A fundamental limitation is computational scalability for large graphs, as matrix exponentials and logarithms remain cubic in $F=U^H$5. However, gradient-based adaptations partly mitigate this constraint.

The AGFRFT framework remains robust and mathematically rigorous for data-driven spectral adaptation and enables joint learning of spectral basis and filter parameters, supporting state-of-the-art denoising and spectral manipulation tasks on graphs (Zhao et al., 20 Nov 2025, Wang et al., 2 Mar 2026).