Multiple-Parameter GFRFT

• MPGFRFT is a generalized framework that applies multiple independent fractional parameters to enable per-frequency control on both undirected and directed graphs.
• It unifies several existing transforms by allowing flexible spectral filtering across multi-dimensional, time-varying, and spatial-temporal signals on irregular domains.
• Its design supports adaptive, learnable spectral filtering and efficient implementations, improving performance in denoising, compression, encryption, and spatiotemporal processing.

The multiple-parameter graph fractional Fourier transform (MPGFRFT) generalizes the classical graph fractional Fourier transform by introducing several independent fractional parameters, enabling a highly flexible fractional spectral analysis of signals supported on graphs, including multi-dimensional, time-varying, and spatial-temporal signals on irregular domains. MPGFRFT unifies and extends the bi-fractional transform on Cartesian product graphs, the angular and joint time-vertex generalizations, and offers multiple construction paradigms for both undirected and directed graphs. This approach underpins adaptive and end-to-end learnable spectral filtering frameworks in modern graph signal processing.

1. Mathematical Formulation and Variants

MPGFRFT enables per-frequency or per-factor fractional spectral control in both undirected and directed graph settings. Let $G=(\mathcal N,\mathcal E,Z)$ be a (possibly directed) graph with diagonalizable shift operator $Z = U\Lambda U^{-1}$.

Single-Parameter GFRFT: For order $a \in \mathbb R$, the classical GFRFT operator is $F^a = V \Lambda_F^a V^{-1}$, with $$\Lambda_F^a = \operatorname{diag}(\mu_0^a, \dots, \mu_{N-1}^a)$$ (Cui et al., 31 Jul 2025).

MPGFRFT–I (Order Vector on Spectrum):

$$F^{\boldsymbol{a}}_{\rm I} = V \operatorname{diag}(\mu_0^{a_0}, \dots, \mu_{N-1}^{a_{N-1}}) V^{-1}$$

with $\boldsymbol{a}=(a_0,\dots,a_{N-1})$ (Cui et al., 31 Jul 2025).

MPGFRFT–II (Non-unitary Polynomial Extension):

$$F^{\boldsymbol{a}}_{\rm II} = \sum_{n=0}^{N-1} C_{n,a_n}^{\rm II} F^n$$

where coefficients mix spectral orders and basis polynomials (non-unitary in general) (Cui et al., 31 Jul 2025).

Multi-dimensional setting (Cartesian product graphs):

Given $d$ undirected graphs $G_i$ ($Z = U\Lambda U^{-1}$0) with orthonormal bases $Z = U\Lambda U^{-1}$1, define the vector of fractional orders $Z = U\Lambda U^{-1}$2. The full MPGFRFT operator on the corresponding Cartesian product is constructed using Kronecker products over the 1D fractional bases: $Z = U\Lambda U^{-1}$3 with each $Z = U\Lambda U^{-1}$4 (Yan et al., 2021, Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).

Dynamic time-vertex domain: The dynamic multiple-parameter joint time-vertex fractional Fourier transform (DMPJFRFT) assigns a vector-valued spatial fractional order at each time index and a time-indexed vector of temporal fractional orders, yielding a block-diagonal/Kronecker construction (Cui et al., 20 Nov 2025).

Directed graphs: For each factor, the SVD of the Laplacian $Z = U\Lambda U^{-1}$5 is used to define fractional powers and associated bases; composition on product graphs proceeds via Kronecker and direct sum constructs (Li et al., 4 Jun 2025).

2. Fundamental Properties

The key algebraic and analytic properties of MPGFRFT derive from its spectral construction and product structure.

• Linearity: MPGFRFT is a linear operator for all variants (Yan et al., 2021).
• Invertibility and Reversibility: Unitary cases (e.g., MPGFRFT–I with symmetric GSO and modulus-one eigenvalues) satisfy

$Z = U\Lambda U^{-1}$6

and similar identity for multi-dimensional MPGFRFT, providing perfect reconstruction (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025, Yan et al., 2021).

• Unitarity: MPGFRFT–I and Cartesian bi-fractional forms are unitary under orthogonal/symmetric eigendecompositions; MPGFRFT–II is not generally unitary (Cui et al., 31 Jul 2025).
• Index Additivity: For order vectors $Z = U\Lambda U^{-1}$7, $Z = U\Lambda U^{-1}$8, type-I satisfies

$Z = U\Lambda U^{-1}$9

and analogously for separable multi-dimensional constructs (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025, Yan et al., 2021).

• Reduction: All schemes collapse to GFRFT or FT for uniform order (all $a \in \mathbb R$0), or to the identity for all orders zero (Wang et al., 13 Oct 2025, Cui et al., 20 Nov 2025, Yan et al., 2021).
• Energy Preservation: Unitary cases inherit Parseval’s equality (Yan et al., 2021, Li et al., 4 Jun 2025).

3. Implementation and Computational Considerations

Efficient MPGFRFT computation leverages separability and spectral decompositions.

• Precomputation: Initial eigendecomposition (or SVD for directed graphs) of the shift operators $a \in \mathbb R$1 or the constituent graphs in product cases (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026, Yan et al., 2021, Li et al., 4 Jun 2025).
• Multi-dimensional applications: Mode-wise application of the transform matrices along each axis yields $a \in \mathbb R$2 complexity, substantially less than full graph diagonalization on the product (Yan et al., 2021).
• Product graph filtering: For a 2D signal on $a \in \mathbb R$3 and $a \in \mathbb R$4:
  • Apply $a \in \mathbb R$5 along rows, $a \in \mathbb R$6 along columns; equivalent to $a \in \mathbb R$7 on the vectorized signal (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).
• Joint time-vertex settings: Block-diagonal in spatial parameter, Kronecker in temporal, for each time slice or vice versa (Cui et al., 20 Nov 2025).
• Parameter selection: Manual sweep, grid search, or end-to-end learnable parameters via backpropagation and matrix differential calculus (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026, Yan et al., 29 Jul 2025).

4. Spectral Filtering and Learnability

MPGFRFT’s parameterization is highly amenable to adaptive filtering and learning frameworks.

• Wiener-style filtering: Diagonal filter design is conducted in the MPGFRFT domain, optimizing for MSE (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).
• End-to-end differentiability: Gradients w.r.t. each parameter (fractional orders, rotation/angle, coupling) are analytically derived, enabling SGD-based optimization of both transform parameters and filter coefficients (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026, Yan et al., 29 Jul 2025).
• Spectral Compression: Custom unitary bases constructed by Gram–Schmidt that align the leading basis vector with the signal achieve near-lossless encoding at ultra-low compression ratios, leveraging the full flexibility of the order vector (Cui et al., 31 Jul 2025).
• Dynamic settings: DMPJFRFT extends this by tailoring the fractional basis to the evolution of the time-vertex structure, further supporting neural network embeddings and gradient-based training (Cui et al., 20 Nov 2025).

5. Relationship to Other Fractional Graph Transforms

MPGFRFT generalizes and subsumes several existing spectral transforms:

Transform	Free Parameters	Description
GFRFT	1 global order	Uniform fractional rotation of spectrum
2D-GFRFT	1 global order (both factors)	Shared-order transform on product graphs
Bi-fractional FRFT	2 (α₁, α₂)	Independent orders for each graph (axes)
AGFRFT	2 (angle θ, order α)	Joint angular and fractional spectral control
JFRFT	2 (α, β)	Fractional order on both vertex and temporal dimensions
DMPJFRFT	$a \in \mathbb R$8 + $a \in \mathbb R$9	Separate per-time and per-frequency fractional orders
MPGFRFT–I/II	N	Per-spectral-mode fractional order

This taxonomy illustrates the progressive increase in modeling capacity, spectral localization, and adaptivity as parameterization becomes richer, with corresponding algorithmic and implementation challenges (Cui et al., 31 Jul 2025, Zhao et al., 20 Nov 2025, Cui et al., 20 Nov 2025, Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).

6. Empirical Outcomes and Applications

Experimental studies consistently show that MPGFRFT and its multi-parameter specializations outperform their single-parameter or fixed-basis counterparts.

• Graph Signal Denoising: MPGFRFT–I/II, especially with learnable order vectors and filters, yields lower MSE and higher SNR compared to GFRFT, GFT, or fixed bases on diverse real datasets (SST, PM2.5, COVID) (Cui et al., 31 Jul 2025, Wang et al., 2 Mar 2026, Wang et al., 13 Oct 2025).
• Image Denoising: Blockwise MPGFRFT–I provides higher PSNR, SSIM, and visually improved output versus classical methods (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025).
• Spectral Compression: Adaptive compression with MPGFRFT achieves relative errors and normalized RMS on the order of $F^a = V \Lambda_F^a V^{-1}$0 and perfect correlation at low retained-coefficient ratios (Cui et al., 31 Jul 2025, Yan et al., 2021).
• Encryption: Use of order vectors as cryptographic keys yields high key sensitivity and drastically expands the key space compared to single-parameter approaches (Cui et al., 31 Jul 2025).
• Spatiotemporal Filtering: The 2D-GBFRFT and geodesic-coupled MPGFRFT consistently surpass traditional 2D-GFRFT and deep learning baselines in dynamic image restoration and time-varying signal denoising, with the hybrid interpolation demonstrating further improvements (Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026).

7. Future Perspectives and Open Issues

• Fast Approximation: High computational complexity (cubic in graph size for naive eigendecomposition) motivates the development of sparsity-aware, approximate, or recursive schemes (Cui et al., 31 Jul 2025).
• Dynamic Graphs: Ongoing development includes smoothness/regularization for order vectors, adaptation to temporally evolving topologies, and further generalization to directed or weighted time-varying graphs (Cui et al., 20 Nov 2025).
• Higher-Dimensional Extensions: The theory and practice of MPGFRFT for products of more than two graphs, fine-grained angular and geodesic parameterizations, and compositions with neural graph learning architectures are emerging directions (Zhao et al., 20 Nov 2025, Cui et al., 20 Nov 2025).
• Control of Non-commutativity: For angular and spectral-deformation extensions, commutation relations between multiple spectral deformation generators complicate invertibility and identifiability, representing both a flexibility and a challenge (Zhao et al., 20 Nov 2025).

MPGFRFT thus stands as a unifying and extensible paradigm for adaptive, learnable, and task-specific spectral analysis of complex graph-supported signals. Its algebraic structure supports robust theoretical properties—linearity, invertibility, unitarity—while its parameterization enables fine-grained and application-driven spectral filtering, with empirical efficacy validated across numerous graph signal processing applications (Cui et al., 31 Jul 2025, Wang et al., 13 Oct 2025, Wang et al., 2 Mar 2026, Zhao et al., 20 Nov 2025, Cui et al., 20 Nov 2025, Yan et al., 2021, Li et al., 4 Jun 2025).