GraphMFT: Fractional Fourier & Fusion

• GraphMFT is a framework that extends the graph Fourier transform using fractional orders to enable adaptive and energy-preserving vertex-frequency analysis.
• It incorporates multi-window techniques that construct tight frames for improved localization, significantly enhancing signal compression, denoising, and anomaly detection.
• GraphMFT also features a graph-based multimodal fusion method that integrates visual, acoustic, and textual data to boost performance in emotion recognition tasks.

GraphMFT refers to two distinct research contributions. In graph signal processing, "GraphMFT" denotes multi-windowed or multiple-parameter fractional Fourier transform techniques on graphs, notably the multi-windowed graph fractional Fourier transform (MWGFRFT) and the multiple-parameter graph fractional Fourier transform (MPGFRFT). Separately, "GraphMFT" also designates a graph-based multimodal fusion technique for emotion recognition in conversation (ERC) utilizing graph neural networks. This article focuses primarily on the advanced fractional Fourier transform methods on graphs, and will also address the ERC-oriented fusion architecture.

1. Foundations: Graph Fractional Fourier Transform and Its Extensions

The classical graph Fourier transform (GFT) leverages the eigenvectors of a graph Laplacian $\mathcal{L}=D-W$ ($D$: degree matrix, $W$: adjacency) for the spectral decomposition of signals defined on graph vertices. The graph fractional Fourier transform (GFRFT) extends this by replacing integer powers with a fractional power $\alpha$: the transform operator $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$, where $U$ diagonalizes $\mathcal{L}$ and $\Lambda$ is the diagonal eigenvalue matrix; $\Lambda^\alpha$ is the diagonal with entries $\lambda_k^\alpha$ ($D$0: eigenvalues) (Shang et al., 2024).

The MWGFRFT further extends this to multi-window settings for improved vertex-frequency localization, drawing inspiration from windowed Fourier and wavelet methods on graphs. Alternatively, the MPGFRFT replaces the global order parameter with a vector $D$1 of per-frequency orders, introducing a unitary operator $D$2 (Cui et al., 31 Jul 2025). Both frameworks offer highly adaptive, invertible, energy-preserving spectral analyses on graphs.

2. Multi-Windowed and Shifted Graph Fractional Fourier Frames

MWGFRFT constructs tight frames by applying multiple analysis windows $D$3 at varying vertex locations and fractional frequencies. Each atom $D$4 is created by fractional translation $D$5 and modulation $D$6 of window $D$7. For a signal $D$8, analysis coefficients are

$D$9

allowing vertex-frequency representations with strong localization. The transform supports exact reconstruction under dual or tight frame conditions, with tightness characterized by uniform partitioning of spectral energy among windows—crucial for stable spectrograms and efficient synthesis (Shang et al., 2024).

The shift multi-windowed variant (SMWGFRFT) introduces a fractional shift operator $W$0 (with $W$1 the adjacency eigendecomposition), and analyzes signals via shifted window atoms, further enhancing time-frequency representation capabilities. Tight-frame and dual-frame properties are established by frame bounds on the spectral content of the shifted windows.

3. Multiple-Parameter Graph Fractional Fourier Transform: Theory and Flexibility

MPGFRFT generalizes GFRFT by assigning an independent order $W$2 to each spectral component, yielding

$W$3

and transformed coefficients $W$4 for signal $W$5. Key properties include unitarity (invertibility with $W$6), additivity ($W$7), and Parseval's theorem ($W$8) (Cui et al., 31 Jul 2025).

A learnable $W$9 enables task-optimized transforms, with gradient-based adjustment for objectives such as sparse energy concentration for compression or denoising. Such optimization is computationally tractable via differentiable spectral operators.

4. Algorithms and Computational Considerations

Efficient computation of MWGFRFT and FMWGFRFT entails precomputing window transforms, fast matrix multiplications with the eigenbasis, and batched inverse operations. Complexity for FMWGFRFT is $\alpha$0, a significant improvement over straightforward $\alpha$1 approaches. For MPGFRFT, each transform/inverse costs $\alpha$2 in dense-eigenbasis regimes, with potential for accelerations via fast Laplacian methods (Shang et al., 2024, Cui et al., 31 Jul 2025).

Key steps include:

• Precomputation of window spectral representations.
• Fast computation of analysis coefficients and their aggregate.
• Inversion for exact synthesis in tight-frame cases, or through dual-frame operators.
• For learnable-order MPGFRFT, iterative gradient steps via backpropagation in the $\alpha$3-vector.

5. Applications: Compression, Denoising, Encryption, and Anomaly Detection

GraphMFT methods provide finely tunable vertex-frequency localization for graph-structured signals. Applications include:

• Spectral Compression: By selecting a compression ratio $\alpha$4, retaining only the largest $\alpha$5 transformed coefficients, perfect reconstruction is possible for $\alpha$6-sparse signals; adaptive $\alpha$7-learning focuses energy for ultra-low ratio compression ($\alpha$8), achieving nearly lossless results on image and climate data (Cui et al., 31 Jul 2025).
• Denoising: Learning per-frequency order vectors for MPGFRFT yields superior denoising efficacy (PSNR $\alpha$9dB, SSIM $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$0 on image patches) relative to single-parameter GFRFT (Cui et al., 31 Jul 2025).
• Encryption: MPGFRFT's large order-vector keyspace dramatically increases security. Combined with DNA-chaos encryption, adjacent-pixel correlations in encoded images drop from $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$1 to near zero, with high sensitivity to key perturbations (Cui et al., 31 Jul 2025).
• Vertex-Frequency Analysis and Anomaly Detection: MWGFRFT and FMWGFRFT outperform classical windowed GFTs for anomaly localization on synthetic and real graphs, with tight frames yielding stable, interpretable spectrograms. The complementarity of FMWGFRFT (extracting main features) and SMWGFRFT (detecting subtle or anomalous components) enables comprehensive analysis (Shang et al., 2024).

Guidelines recommend choosing window type (e.g., heat kernel, B-spline) for time–vertex trade-offs and tuning $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$2 to emphasize desired localization. Tight-frame constructions simplify reconstructions and enhance interpretability.

6. GraphMFT for Multimodal Emotion Recognition in Conversation

A separate branch of research introduces GraphMFT as a graph network-based multimodal fusion technique for ERC (Li et al., 2022). Here, GraphMFT constructs three heterogeneous graphs (V-A, V-T, A-T) to facilitate information exchange between visual, acoustic, and textual modalities using improved graph attention networks (GATs). By learning sparser, context-aware intra-/inter-modal edge weights and combining modality-specific updates, GraphMFT overcomes over-smoothing and noisy edge issues present in prior graph-based fusion models. Experimental results on IEMOCAP and MELD datasets establish state-of-the-art performance (accuracy $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$3 and $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$4, respectively), with ablation studies underscoring the benefits of the improved GAT and two-modality graph structure.

7. Limitations and Future Directions

For signal processing, residual challenges include nontrivial parameter selection (window type, $F^\alpha = U\Lambda^\alpha U^\mathsf{T}$5 tuning), scaling of eigenbasis computations, and potential adaptation to dynamic or directed graphs. Open extensions involve integrating external or task-dependent graph structures, applying GraphMFT to broader classes of signals (multimodal, temporal, or knowledge-graph data), and further optimizing transforms for compressed sensing and privacy-preserving analytics (Shang et al., 2024, Cui et al., 31 Jul 2025).

For multimodal ERC, future work points to extending the fusion framework to additional modalities (gestures, physiological signals), dynamic graph construction (learning edge existence and weights), incorporating external knowledge graphs, and transplanting the architecture to tasks such as sentiment analysis and video QA.

---

References:

• Frames and vertex-frequency representations in graph fractional Fourier domain (Shang et al., 2024)
• Multiple-Parameter Graph Fractional Fourier Transform: Theory and Applications (Cui et al., 31 Jul 2025)
• GraphMFT: A Graph Network based Multimodal Fusion Technique for Emotion Recognition in Conversation (Li et al., 2022)