Generalized Fractional Filtering Embedding (GEFRFE)

• GEFRFE is a graph representation learning framework that extends classical spectral embeddings into the fractional domain using nonlinear eigenvector compositions.
• It applies the graph fractional Fourier transform and dynamic selection of the fractional parameter α to optimize filtering and improve classification accuracy without added complexity.
• Empirical results on benchmark graph datasets demonstrate that GEFRFE consistently outperforms traditional methods, offering practical benefits in real-world graph learning tasks.

The Generalized Fractional Filtering Embedding (GEFRFE) is a graph representation learning framework that extends classical spectral graph embedding into the fractional transform domain. By leveraging the graph fractional Fourier transform (GFRFT) and nonlinear combinations of fractional Laplacian eigenvectors, GEFRFE generates expressive, low-dimensional graph representations that capture structural features invisible in the standard spectral (α=1) case. The approach enables dynamic selection of the fractional parameter α through both exhaustive search and adaptive learning within deep neural networks, resulting in richer embeddings and superior classification accuracy without increasing asymptotic computational complexity compared to prior spectral methods (Sheng et al., 4 Aug 2025).

1. Fractional Laplacian and Graph Fractional Fourier Transform

For a simple undirected graph $G=(V,A)$ with $N$ nodes, degree matrix $D$, and Laplacian $L = D - A$, the spectrum is obtained from the eigendecomposition $L = U \Lambda U^\top$, where $U$ is orthonormal and $\Lambda$ is diagonal with nonnegative eigenvalues. The fractional power of the Laplacian is defined by

$$L^\alpha = U \Lambda^\alpha U^\top\,,\quad \Lambda^\alpha = \mathrm{diag}(\lambda_1^\alpha, \ldots, \lambda_N^\alpha)\,,\quad \alpha \in \mathbb{R}\,.$$

The GFRFT matrix is constructed as

$F^\alpha = U\,\Lambda^{\alpha/2}\,U^\top\,,$

where the action of $F^\alpha$ on a graph signal $N$0 yields the fractional spectrum $N$1, with fractional eigenvectors $N$2 (columns of $N$3). For $N$4, GFRFT reduces to the standard graph Fourier transform (GFT); for $N$5, it reverts to the identity map (vertex domain) (Sheng et al., 4 Aug 2025, Yan et al., 29 Jul 2025).

2. Fractional Filtering Embedding Operator

GEFRFE generalizes the spectral filtering embedding paradigm by combining a bank of $N$6 diagonal spectral filters $N$7 and a set of $N$8 power-orders $N$9. The embedding construction for a constant input signal proceeds as follows:

1. Compute fractional GFT eigenvectors: $D$0.
2. Form power-sum features: For each eigenvector, compute

$D$1

for $D$2.

1. Apply each filter $D$3 to generate filtered responses:

$D$4

2. The final GEFRFE embedding is the $D$5 matrix

$D$6

stacked over all filters and power-orders.

When $D$7, this process exactly reproduces the Generalized Frequency Filtering Embedding (GEFFE) (Sheng et al., 4 Aug 2025).

3. Nonlinear Eigenvector Component Compositions

The core of GEFRFE's representational power is the nonlinear composition of fractional GFT eigenvectors via power-summation and filter-weighting. The fractional parameter $D$8 modulates the localization and mixing of spectral components, yielding feature vectors that are sensitive to graph structures not detectable in the $D$9 pure spectral domain. Varying the power-order $L = D - A$0 and filter $L = D - A$1 allows the uncovering of complementary structural signatures from the fractional spectrum, enhancing the expressiveness and informativeness of the embedding (Sheng et al., 4 Aug 2025).

4. Strategies for Dynamic Selection of $L = D - A$2

Two parallel dynamic selection strategies for the fractional parameter $L = D - A$3 are used to maximize the informativeness of the embedding:

• Grid Search Optimization: The space $L = D - A$4 is discretized ($L = D - A$5) and, for each $L = D - A$6, the GEFRFE embedding is computed and used for $L = D - A$7-nearest neighbor graph classification. The optimal $L = D - A$8 is chosen to maximize averaged accuracy over multiple random splits (20 in typical experimental setups).
• Adaptive Learning via ResNet-18: The embedding $L = D - A$9 is passed to a ResNet-18 classifier. Here, $L = U \Lambda U^\top$0 is treated as a learnable parameter, updated via gradient backpropagation alongside network weights under a cross-entropy loss with Adam optimization and scheduling. Control experiments with fixed $L = U \Lambda U^\top$1 verify that performance gains stem from fractionalization, not increased model capacity (Sheng et al., 4 Aug 2025).

5. Computational Complexity

The dominant computation for both GEFFE and GEFRFE is the Laplacian eigendecomposition ($L = U \Lambda U^\top$2). Other steps—construction of $L = U \Lambda U^\top$3, fractional transform, filtering—are $L = U \Lambda U^\top$4 or $L = U \Lambda U^\top$5. Therefore, GEFRFE retains the overall $L = U \Lambda U^\top$6 complexity of GEFFE. Additional overhead in GEFRFE arises from scanning multiple $L = U \Lambda U^\top$7 values (grid search) or extended training epochs (when learning $L = U \Lambda U^\top$8), but does not alter the fundamental asymptotic cost (Sheng et al., 4 Aug 2025).

Method	Eigendecomp	Transform	Filtering	Total Complexity
GEFFE	$L = U \Lambda U^\top$9	$U$0	$U$1	$U$2
GEFRFE	$U$3	$U$4	$U$5	$U$6

6. Experimental Performance and Empirical Insights

Extensive evaluation on benchmark graph datasets—including Llow, Lmed, Lhigh (each: 2250 graphs, 15 classes), PROTEINS (1113 graphs, 2 classes), IMDB-MULTI (1500 graphs, 3 classes), and NCI1 (4110 graphs, 2 classes)—demonstrates that GEFRFE with optimally selected or adaptively learned $U$7 consistently outperforms GEFFE.

Key findings:

• On Llow, accuracy increases from 48.23% (GEFFE) to 66.90% (GEFRFE, best $U$8).
• On Lmed, from 41.68% to 58.18%.
• On PROTEINS, from 69.51% to 71.11%.
• Using ResNet-18 with learnable $U$9, peak accuracies of 86.89% (Llow), 78.00% (Lmed), 59.78% (Lhigh), 69.14% (NCI1), 72.65% (PROTEINS), and 46.33% (IMDB-MULTI) are observed.
• Different filter types (heat kernels, anti-heat, part-sine, polynomial) and power-orders $\Lambda$0 reveal additional structure, confirming the complementarity provided by the fractional domain embedding (Sheng et al., 4 Aug 2025).

The framework is thus empirically validated to yield both more informative embeddings and significant accuracy improvements in graph classification tasks, while maintaining practical computational efficiency.

7. Extensions and Related Fractional Embedding Paradigms

The inclusion of fractional orders in transform-based embedding aligns with trends in time-vertex signal processing, such as the Generalized Fractional Filtering Embedding (GE-FRFE) for spatiotemporal graph signals. Here, hyper-differential forms of the GFRFT and the joint time-vertex fractional Fourier transform (JFRFT) enable adaptive transform order (and spectral filter) learning directly via backpropagation in neural architectures, offering both enhanced denoising and reduced computational burden compared to grid search strategies (Yan et al., 29 Jul 2025). This suggests a unifying theoretical substrate for fractional-domain representation learning across static and dynamic graphs.