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arxiv: 2508.02383 · v2 · pith:5HKGIHY3new · submitted 2025-08-04 · 💻 cs.LG · cs.IR

Graph Embedding in the Graph Fractional Fourier Transform Domain

Changjie Sheng , Zhichao Zhang , Yangfan He This is my paper

Pith reviewed 2026-05-21 23:09 UTC · model grok-4.3

classification 💻 cs.LG cs.IR
keywords graph embeddinggraph fractional Fourier transformspectral embeddinggraph representation learningfractional domainnode classificationgraph Laplacian
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The pith

Graph embedding in the fractional Fourier transform domain captures richer structural features and improves classification on benchmark datasets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the generalized fractional filtering embedding (GEFRFE) to extend traditional spectral graph embedding into fractional domains using the graph fractional Fourier transform. This approach uses graph fractional domain filtering along with a nonlinear composition of components from a fractionalized graph Laplacian to make embeddings more informative. Fractional orders are selected through search-based optimization or ResNet18 adaptive learning. Tests on five standard datasets indicate better capture of latent graph structures and higher classification accuracy. If correct, this shifts graph embedding methods toward more flexible, generalized transform domains while keeping computation similar to prior methods.

Core claim

The GEFRFE extends the generalized frequency filtering embedding into fractional domains by applying the graph fractional Fourier transform. It enhances informativeness through fractional domain filtering and nonlinear composition of eigenvector components from the fractionalized graph Laplacian. Two strategies dynamically set the fractional order: search-based optimization and ResNet18-based adaptive learning. Experiments confirm richer structural features and significantly better classification performance on five benchmark datasets, establishing a path from fixed to generalized domains in graph embedding.

What carries the argument

The graph fractional Fourier transform applied via filtering in the fractional domain combined with nonlinear composition of eigenvector components from a fractionalized graph Laplacian.

Load-bearing premise

The two strategies for selecting the fractional order will consistently yield embeddings that generalize well across datasets without overfitting or high extra cost.

What would settle it

If applying the GEFRFE to the five benchmark datasets yields classification performance no better than or worse than the non-fractional GEFFE method, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2508.02383 by Changjie Sheng, Yangfan He, Zhichao Zhang.

Figure 1
Figure 1. Figure 1: The structural information of a graph in different embedding domains. (a): Vertex domain. (b): Graph spectral domain embedding, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Visualization schematic diagram of the embedded feature matrix of multiple graph signals. The rows of the matrix represent different graph [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The flowchart of embedding. The upper part of the figure illustrates the existing GEFFE method, whose main limitation lies in performing embeddings [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The proposed filter function sets. (a): H, AH, PS filters. (b): X filter. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The classification accuracies of GEFRFE and GEFFE. (a): GEFRFE: [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Filter combinations with three different combination strategies. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The classification accuracies of adaptive learning and forward [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Spectral graph embedding plays a critical role in graph representation learning by generating low-dimensional vector representations from graph spectral information. However, the embedding space of traditional spectral embedding methods often exhibit limited expressiveness, failing to exhaustively capture latent structural features across alternative transform domains. To address this issue, we use the graph fractional Fourier transform to extend the existing state-of-the-art generalized frequency filtering embedding (GEFFE) into fractional domains, giving birth to the generalized fractional filtering embedding (GEFRFE), which enhances embedding informativeness via the graph fractional domain.The GEFRFE leverages graph fractional domain filtering and a nonlinear composition of eigenvector components derived from a fractionalized graph Laplacian. To dynamically determine the fractional order, two parallel strategies are introduced: search-based optimization and a ResNet18-based adaptive learning. Extensive experiments on five benchmark datasets demonstrate that the GEFRFE captures richer structural features and significantly enhance classification performance. The GEFRFE provides a new paradigm for the development of graph embedding from the "fixed domain" to the "generalized domain". The results indicate that introducing the GFRFT into the graph embedding domain is a correct and effective research path. Notably, the proposed method retains computational complexity comparable to GEFFE approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes extending the generalized frequency filtering embedding (GEFFE) into the graph fractional Fourier transform domain to create the generalized fractional filtering embedding (GEFRFE). It applies graph fractional domain filtering and a nonlinear composition of eigenvector components derived from a fractionalized graph Laplacian. Two strategies are introduced for selecting the fractional order: search-based optimization and ResNet18-based adaptive learning. The central claim is that GEFRFE captures richer structural features than standard spectral methods and significantly improves classification performance on five benchmark datasets while retaining computational complexity comparable to GEFFE.

Significance. If the performance improvements can be isolated from hyperparameter effects, the introduction of GFRFT-based filtering and nonlinear eigenvector composition could meaningfully extend spectral graph embedding techniques by enabling exploration of generalized transform domains. The dual strategies for fractional-order selection and the claim of comparable complexity are potentially useful contributions, but the significance hinges on demonstrating that the fractional approach adds expressiveness beyond what equivalent tuning achieves in non-fractional baselines.

major comments (2)
  1. The abstract and experimental claims assert significant performance gains from the fractional approach and nonlinear composition but supply no quantitative metrics, error bars, baseline comparisons, or methodological details on fractional Laplacian construction and ResNet18 integration. This absence is load-bearing for evaluating whether the reported improvements on the five benchmarks are reliable.
  2. The search-based optimization strategy for the fractional order effectively introduces a continuous hyperparameter tuned per dataset. Without an ablation that fixes the order at 1 (recovering GEFFE) or compares against a non-fractional baseline given equivalent tuning effort, it remains unclear whether classification gains arise from the GFRFT and nonlinear composition or from the additional optimization freedom. This directly affects the central claim that the method yields richer embeddings due to the fractional domain.
minor comments (2)
  1. Clarify the precise mathematical definition of the nonlinear composition of fractional eigenvector components and how it differs from standard spectral operations.
  2. Provide explicit pseudocode or complexity analysis to support the claim that computational complexity remains comparable to GEFFE.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, providing honest responses and committing to revisions that strengthen the experimental rigor and clarity without misrepresenting our contributions.

read point-by-point responses

  1. Referee: The abstract and experimental claims assert significant performance gains from the fractional approach and nonlinear composition but supply no quantitative metrics, error bars, baseline comparisons, or methodological details on fractional Laplacian construction and ResNet18 integration. This absence is load-bearing for evaluating whether the reported improvements on the five benchmarks are reliable.

    Authors: We agree that the abstract would benefit from greater specificity to support the performance claims. While the full manuscript (Section 4) reports classification accuracies, standard deviations across repeated trials, and comparisons against baselines including GEFFE on the five datasets, the abstract summarizes these results qualitatively. In the revised version, we will update the abstract to include representative quantitative metrics (e.g., average accuracy gains) and explicitly reference the use of error bars. Methodological details on the fractional Laplacian (constructed via eigendecomposition raised to fractional power) and ResNet18-based adaptive order selection are provided in Sections 3.2 and 3.3; we will add a concise summary of these elements to the abstract or introduction to improve self-containment and evaluability. revision: yes

  2. Referee: The search-based optimization strategy for the fractional order effectively introduces a continuous hyperparameter tuned per dataset. Without an ablation that fixes the order at 1 (recovering GEFFE) or compares against a non-fractional baseline given equivalent tuning effort, it remains unclear whether classification gains arise from the GFRFT and nonlinear composition or from the additional optimization freedom. This directly affects the central claim that the method yields richer embeddings due to the fractional domain.

    Authors: This concern is valid and directly impacts the interpretation of our central claim. The search-based strategy does introduce additional optimization freedom for the fractional order. To isolate the contribution of the GFRFT and nonlinear eigenvector composition, we will add ablation experiments in the revised manuscript: (i) fixing the order at 1 to recover GEFFE, (ii) comparing against non-fractional baselines given equivalent hyperparameter tuning budget, and (iii) contrasting the search-based and ResNet18 adaptive strategies. These controls will clarify whether gains arise from the fractional domain itself. We believe the results will support our claim, but we acknowledge the current presentation requires these additions for rigor. revision: yes

Circularity Check

1 steps flagged

Fractional order selection via optimization or learning may account for gains without isolating the GFRFT contribution

specific steps
  1. fitted input called prediction [Abstract]
    "To dynamically determine the fractional order, two parallel strategies are introduced: search-based optimization and a ResNet18-based adaptive learning. Extensive experiments on five benchmark datasets demonstrate that the GEFRFE captures richer structural features and significantly enhance classification performance."

    The order parameter is tuned or learned on the target data to produce the reported embeddings and classification results; the claimed richer features and performance gains are then presented as evidence for the GEFRFE method itself, without an independent validation that separates the transform properties from the optimization outcome.

full rationale

The derivation extends GEFFE via GFRFT and nonlinear eigenvector composition, with fractional order chosen by search optimization or ResNet18 adaptation. Performance claims on benchmarks follow directly from these choices, but no explicit reduction to input data by construction is shown in the provided text. The adaptive strategies introduce tunable parameters whose effect is not ablated against fixed-order baselines, creating partial circularity risk in attributing improvements to the fractional domain rather than extra fitting freedom. This matches a fitted-input pattern but remains non-load-bearing for the core extension claim, which retains independent content from the transform definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that fractional Fourier transforms can be meaningfully defined and applied to graphs via a fractionalized Laplacian, plus the practical introduction of optimization and neural adaptive strategies for the fractional order without first-principles derivation.

free parameters (1)
  • fractional order
    Dynamically selected via search optimization or ResNet18 learning to control the transform domain; acts as a tunable parameter central to the embedding construction.
axioms (1)
  • domain assumption The graph Laplacian admits a meaningful fractionalization that preserves useful spectral properties for embedding.
    Invoked when defining the fractionalized graph Laplacian and applying GFRFT for eigenvector components.
invented entities (1)
  • nonlinear composition of fractional eigenvector components no independent evidence
    purpose: To enhance embedding informativeness beyond linear spectral filtering.
    New mechanism introduced in the GEFRFE construction without independent evidence outside the proposed method.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. FGFRFT: Fast Graph Fractional Fourier Transform via Exact Spectral Splitting and Fourier-Series Approximation

    eess.SP 2026-02 unverdicted novelty 6.0

    FGFRFT splits the spectrum of a unitary GFT to treat λ=-1 exactly and approximates the complementary part by a length-L Fourier series, reducing online complexity to O(2 L N²) with derived error bounds.

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