Khan-GCL: Kolmogorov-Arnold Network Based Graph Contrastive Learning with Hard Negatives

Boxun Xu; Hejia Geng; Peng Li; Zihu Wang

REVIEW 3 major objections 6 minor 60 references

Khan-GCL: Kolmogorov-Arnold Network Based Graph Contrastive Learning with Hard Negatives

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Khan-GCL claims that coupling a Kolmogorov-Arnold encoder with hard negatives mined from B-spline coefficients makes graph contrastive learning transfer and classify better.

desk verdict A new hard-negative scheme for KAN-based graph contrastive learning with a plausible empirical case, but the variance-bound theorem behind the discriminative score is false and the baseline comparisons need a common codebase. read the letter →

arxiv 2505.15103 v2 pith:YFNKV2TJ submitted 2025-05-21 cs.LG

Zihu Wang , Boxun Xu , Hejia Geng , Peng Li This is my paper

classification cs.LG

keywords graphcontrastivelearningKolmogorov-ArnoldnetworkshardnegativesB-splinecoefficientscriticalfeatureidentificationself-supervisedrepresentationclassificationtensordecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that graph contrastive learning can be improved on two fronts at once: the encoder and the negatives. It replaces the usual MLP encoder with a Kolmogorov-Arnold Network (KAN), whose learnable B-spline activations add expressive capacity without going deeper, and it mines the KAN's coefficients to identify which representation dimensions are most independent and most discriminative. Perturbing precisely those dimensions produces hard negatives—examples that stay close to the original graph but push the encoder to separate different classes. If the claim holds, label-free pre-training on unlabeled graphs yields representations that transfer and classify better, which is why the approach is aimed at domains like molecular property prediction and social network analysis.

What carries the argument

The load-bearing object is the KAN layer's coefficient tensor $C = \{c_{ijk}\} \in \mathbb{R}^{d_{\mathrm{in}} \times d_{\mathrm{out}} \times d_c}$, where each univariate activation is a B-spline $\phi_{i,j}(x)=\sum_k c_{ijk}B_{ijk}(x)$. CKFI reads this tensor twice: a leave-one-dimension-out HOSVD reconstruction error $\delta_j$ scores how independently each output dimension encodes information, and the average coefficient variance $\rho_j$ scores how much that dimension spreads the data. A hard negative is the representation $z$ plus $p_\delta+p_\rho$, where each coordinate perturbation is a Rademacher-signed Gaussian with mean proportional to $\delta_j$ or $\rho_j$, and a stop-gradient on the projected hard negatives in the extra loss term prevents collapse. The mechanism carries the paper's claim that perturbing the most independent and most discriminative dimensions changes a graph's semantic identity with only a small feature-space displacement.

What would settle it

Check the bound numerically for a uniform cubic B-spline with slowly varying coefficients, e.g., $c_k = 1 + 0.01\sin(k\pi/n)$, computing $\mathrm{Var}[\phi(x)] = d^T M d$ with $d_i = c_i - \bar{c}$ and comparing it to $M^{(0)}\sigma_c^2$; with positive off-diagonal overlap integrals $M^{(1)}, M^{(2)}, M^{(3)}$, this quantity can exceed the claimed upper bound, which would refute the proposition as stated.

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Extended reading notes

Core claim

Khan-GCL's central claim is that the coefficient tensor of a KAN encoder carries global information that can be turned into a training signal for contrastive learning. The paper derives two scores for each output dimension: an independence score $\delta_j$, defined as the Frobenius reconstruction error when that dimension's coefficient slice is omitted from a higher-order singular value decomposition (HOSVD), and a discriminative score $\rho_j$, the average variance of the B-spline coefficients feeding that dimension. It then forms hard negatives by adding small Rademacher-signed Gaussian perturbations whose means are proportional to these scores, and trains with the standard contrastive loss plus a stop-gradient hard-negative loss. In the reported experiments this combination attains the best average results on eight biochemical transfer-learning datasets and eight TU unsupervised datasets, and the MNIST-superpixel nearest-neighbor retrieval shows that the generated hard negatives are visually similar to the original graphs yet mostly carry different labels.

Load-bearing premise

The load-bearing premise is that an output dimension with high variance among its spline coefficients is genuinely more discriminative; the paper's proof of that bound treats all overlap integrals as nonnegative, which need not hold when the coefficients vary slowly.

Editorial extensions

If this is right

Transfer learning results on MoleculeNet datasets improve when the pre-trained encoder is a KAN: Khan-GCL reports the best average ROC-AUC across eight biochemical benchmarks compared with the listed GCL methods.
On eight TU-datasets under the SVM evaluation protocol, Khan-GCL reports the highest average accuracy, with the best single-dataset accuracy on six of the eight.
Replacing MLP encoders with same-sized KAN layers improves GraphCL and JOAOv2 even before hard negatives are added, indicating the encoder change is beneficial on its own.
Hard negatives generated by CKFI outperform random Gaussian perturbation, and using both the independent and discriminative scores together outperforms using either type alone.
The added computational cost of KAN spline evaluation is moderate: in Zinc-2M pre-training, Khan-GCL is roughly 0.063 seconds per iteration, close to the RGCL baseline and about 1.4 times the GraphCL baseline.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The CKFI score computation assumes that the coefficient tensor's mode-2 slices are semantically meaningful; if that holds, the same scheme could be applied to intermediate KAN layers to generate layer-wise hard negatives, which the authors list only as future work.
Since the construction needs only a coefficient tensor, it should carry over to other spline- or wavelet-based networks and to tensorized MLPs, making targeted hard-negative generation a property of the parameterization rather than a graph-specific trick.
One direct test of the semantic claim: on a labeled dataset, perturbing only the top-$\delta$ and top-$\rho$ dimensions should flip a nearest-neighbor or linear-classifier label more often than perturbing the same number of random dimensions at equal average magnitude.
Pre-trained CKFI scores could be frozen and reused when fine-tuning on small downstream datasets, offering a label-free form of hard-negative augmentation during adaptation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid empirical paper with a broken theorem. The KAN encoder is not shocking, but CKFI — using HOSVD leave-one-out reconstruction error to find independent output dimensions and B-spline coefficient variance to find discriminative ones — is a genuinely new mechanism, and the experiments are wide enough to take the empirical claim seriously. It deserves peer review, but the referee should require the Proposition 2 proof to be fixed or removed before acceptance.

What's actually new: as far as I can tell, this is the first GCL method that uses a KAN encoder and mines hard negatives by perturbing the encoder's output dimensions in proportion to a coefficient-derived importance score. The two scoring ideas are distinct and interesting. The independence score from tensor decomposition is new to GCL, and reading coefficient variance as a global proxy for output variance is a reasonable heuristic even if the formal bound fails.

The experimental work is above the median for this area. They run transfer learning on 8 MoleculeNet datasets and unsupervised pre-training on 8 TU datasets, report standard deviations, include ablations that separate the contribution of the KAN from the hard negatives, show both perturbation types help and random perturbation doesn't, and report runtime overhead that is modest. The MNIST-superpixel nearest-neighbor figure is a nice qualitative check that the generated negatives are hard rather than random noise.

Where it gets shaky: Proposition 2 is false as stated. The proof in Appendix A.2 expands Var[phi] = d^T M d and then drops all the off-diagonal 2M(k) sum d_i d_{i+k} terms because M(k) >= 0. Nonnegative entries don't imply those quadratic terms are nonnegative; for a low-frequency zero-mean coefficient vector, the off-diagonal contribution is positive, and the bound M(0) sigma_c^2 is exceeded. So the formal justification for the discriminative score is unsupported. The heuristic may still work — the ablation says it does — but the theorem needs to be corrected, specialized, or retracted. This is the main correctness issue.

Two more soft spots, both in the "standard for the field but still a weakness" category: baseline numbers are quoted from original papers rather than rerun in one codebase, and no code is released. The HOSVD truncation ranks are also not disclosed, so the independence score isn't reproducible.

Bottom line: the empirical central claim is credible, but the paper oversells itself by resting the discriminative feature on a false inequality. A serious referee should engage: the work is worth revising, and the experimental parts are valuable on their own. I'd send it to peer review with a clear request for proof repairs, code, and baseline reruns.

Referee Report

3 major / 6 minor

Summary. The paper proposes Khan-GCL, a graph contrastive learning framework that replaces the MLP encoder with a KAN-based encoder and adds a hard-negative generation module (CKFI). CKFI identifies two types of critical output dimensions—'independent' dimensions via HOSVD reconstruction error and 'discriminative' dimensions via the variance of B-spline coefficients—and perturbs those dimensions in representation space to create hard negatives. The authors report state-of-the-art transfer learning and unsupervised graph classification results on MoleculeNet and TU-dataset benchmarks, with ablations isolating the KAN encoder and the two CKFI components, plus a qualitative MNIST-superpixel retrieval study.

Significance. If the empirical results are taken at face value, the paper makes a useful practical contribution: it demonstrates that KAN encoders can be plugged into GCL pipelines and that coefficient-based feature scoring is a cheap alternative to batch-wise variance estimation. The ablations (Table 4, Figure 4) and the compatibility experiment with JOAOv2 are informative, and the MNIST-superpixel nearest-neighbor check is a nice sanity test. However, the theoretical justification for the discriminative component is a named contribution and is currently invalid; the SOTA claim also rests on quoted baselines rather than a controlled rerun. With those points repaired, the empirical study would be a solid contribution to the GCL/KAN line of work.

major comments (3)

[Section 4.2, Proposition 2 and Appendix A.2 (Eqs. 26–27)] The proof of Proposition 2 is invalid. In Eq. (26), the off-diagonal terms 2M(k) Σ d_i d_{i+k} are not nonnegative in general: for a zero-mean, slowly varying coefficient deviation such as d_i = cos(2π i/n) after centering, the products d_i d_{i+1} are predominantly positive, so 2M(1) Σ d_i d_{i+1} > 0 and the claimed inequality Var[ϕ] ≤ M(0) Σ d_i^2 does not follow. Equation (27) also omits the factor n when replacing Σ d_i^2 by σ_c^2, since Σ d_i^2 = n σ_c^2. Because Eq. (8) and Eq. (9) use coefficient variance as the discriminative score, the stated theoretical justification for this part of CKFI is unsupported. The authors should either prove a correct bound (e.g., using λ_max(M)) and adapt the score accordingly, or explicitly reframe the discriminative score as a heuristic.
[Section 4.2–4.3] Even if Proposition 2 were true for a single univariate B-spline, it would not establish Eq. (8)'s ρ_j as a measure of output-dimension discriminative power, because the KAN layer output in Eq. (2) is a sum of d_in such functions and the variance of that sum depends on the input distributions and cross terms between the univariate functions. The paper should either derive a valid bound for the layer output variance or present CKFI as a heuristic motivated by the coefficient-variance intuition.
[Section 5, Tables 1–3] All comparison numbers for prior methods are quoted from their original publications rather than reproduced in a shared pipeline, as the table footnotes acknowledge. This weakens the central 'state-of-the-art' claim: differences in hyperparameters, dataset splits, and evaluation details can change ROC-AUC or accuracy by more than the reported margins. I ask the authors to rerun the most competitive baselines under their own evaluation protocol, or to substantially soften the SOTA claim.

minor comments (6)

[Appendix C, Algorithm 2] The loop header says 'for i = 1 to dc' but the body removes slice j and computes δ_j; the loop should iterate over output dimensions j.
[Appendix E.1] The dataset list names only seven MoleculeNet datasets (BBBP, Tox21, SIDER, ClinTox, MUV, HIV, BACE), but Table 1 also includes ToxCast; the list should be completed.
[Tables 1 and 2] The column header 'A VG' appears to be a typo for 'Avg'.
[Section 1, Introduction] The phrase 'balance between expensiveness and risks' should read 'balance between expressiveness and risks'.
[Appendix B] The heading contains the typo 'Algorothm' and should read 'Algorithm'.
[Reproducibility] The paper does not state whether code will be released; given the nontrivial implementation details in Algorithms 1–3, a code release or pseudocode-level clarification of the least-squares projection in Algorithm 2 (line 6) would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Khan-GCL's empirical claims are benchmarked against externally published results, and CKFI's self-referential coefficient scoring is standard hard-negative mining rather than a derivation that reduces to its input.

full rationale

No load-bearing circular step was found. The central claims are empirical: replacing MLP encoders with KAN encoders and using CKFI-based perturbations to generate hard negatives, with performance compared against independently published results in Tables 1-3 and ablations in Table 4 and Figure 4. CKFI computes discriminative and independence scores from the current KAN coefficients and uses those scores to perturb representations for an additional contrastive loss; this is a self-referential training loop common to hard-negative mining, not a derivation whose conclusion is assumed by construction. Proposition 2's variance bound is offered as mathematical support, and the Appendix A.2 proof is questionable because it drops off-diagonal quadratic terms that need not be nonnegative, but an unsupported or false supporting bound is a correctness risk, not circularity, since the empirical method does not reduce to that bound. Self-citations in the introduction are contextual and not load-bearing for the main result. The paper is self-contained against external benchmarks, so the circularity score is 0.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

No invented physical entities are introduced. The main load-bearing pieces are the CKFI scores: the independent score depends on an unspecified HOSVD rank selection, and the discriminative score depends on the unproved variance bound from Proposition 2. The perturbation hyperparameters are hand-set and tuned, though the same values are used across datasets.

free parameters (7)

epsilon_delta = 0.075
Scales the mean of the Gaussian perturbation applied to independent dimensions (Eq 9); tuned for both transfer and unsupervised settings (Appendices F.2, F.3).
epsilon_rho = 0.075
Scales the mean of the Gaussian perturbation applied to discriminative dimensions (Eq 9); tuned in both settings.
sigma_delta = 0.05
Standard deviation of the perturbation noise for independent dimensions (Eq 9).
sigma_rho = 0.05
Standard deviation of the perturbation noise for discriminative dimensions (Eq 9).
HOSVD truncation ranks (r1, r2, r3) = unspecified
Algorithm 1 truncates U(i) to r_i but the paper never states how these ranks are chosen; they control the reconstruction error scores delta_j (Eq 6).
KAN grid size = 5
Number of B-spline intervals in each KAN activation (Appendix F.1); affects the coefficient tensor C and all CKFI scores.
B-spline order = 3 (cubic)
Cubic B-splines are used throughout; the variance bound in Proposition 2 is derived specifically for cubic splines (Appendix A.2).

assumptions (5)

standard math Kolmogorov-Arnold representation theorem justifies KAN encoders
The paper builds on Liu et al. 2024, treating KAN's expressive power as given; not derived in this paper.
domain assumption Uniform B-spline basis functions satisfy partition of unity and have equal integrals
Appendix A.2 assumes all B-spline basis integrals omega_k are equal and sum to b-a; this holds for periodic uniform B-splines but not necessarily for clamped boundary conditions used in practice.
ad hoc to paper Variance of a uniform B-spline function is bounded by M(0) times coefficient variance
Proposition 2 is the foundation for the discriminative score rho_j; the proof in Appendix A.2 Eq (27) drops positive cross terms, so the bound is not established and is questionable for slowly varying coefficient vectors.
ad hoc to paper Leave-one-out HOSVD reconstruction error identifies semantically critical output dimensions
No theoretical or empirical validation beyond the UMAP figure connects reconstruction error delta_j to semantic importance; this is a heuristic.
ad hoc to paper Perturbing high-scoring dimensions changes graph identity while preserving similarity
The hard negative generation (Eq 9-10) assumes that dimensions with large coefficient variance or reconstruction error are the ones that, when perturbed, produce a different class with similar appearance; the nearest neighbor visualization is suggestive but not a controlled test.

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Cite this review

Pith. "Pith review of Khan-GCL: Kolmogorov-Arnold Network Based Graph Contrastive Learning with Hard Negatives." pith.science (2026). https://pith.science/paper/YFNKV2TJ

@misc{pith2026250515103,
  author       = {Pith},
  title        = {Pith review of: Khan-GCL: Kolmogorov-Arnold Network Based Graph Contrastive Learning with Hard Negatives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFNKV2TJ}},
  note         = {Machine review of arXiv:2505.15103}
}

read the original abstract

Graph contrastive learning (GCL) has demonstrated great promise for learning generalizable graph representations from unlabeled data. However, conventional GCL approaches face two critical limitations: (1) the restricted expressive capacity of multilayer perceptron (MLP) based encoders, and (2) suboptimal negative samples that either from random augmentations-failing to provide effective 'hard negatives'-or generated hard negatives without addressing the semantic distinctions crucial for discriminating graph data. To this end, we propose Khan-GCL, a novel framework that integrates the Kolmogorov-Arnold Network (KAN) into the GCL encoder architecture, substantially enhancing its representational capacity. Furthermore, we exploit the rich information embedded within KAN coefficient parameters to develop two novel critical feature identification techniques that enable the generation of semantically meaningful hard negative samples for each graph representation. These strategically constructed hard negatives guide the encoder to learn more discriminative features by emphasizing critical semantic differences between graphs. Extensive experiments demonstrate that our approach achieves state-of-the-art performance compared to existing GCL methods across a variety of datasets and tasks.

Figures

Figures reproduced from arXiv: 2505.15103 by the authors.

**Figure 1.** Left: Overview of the Khan-GCL framework. The encoder utilizes KAN to enhance expressive power and interpretability. Leveraging the KAN architecture, we introduce two critical dimension identification techniques. By applying small perturbations to these identified dimensions, we generate hard negative samples for each graph, thereby improving the performance of this GCL approach. Right: UMAP(2018) visualization of … view at source ↗

**Figure 2.** Illustration of a KAN layer and our two proposed critical feature identification techniques. (a) An output dimension is [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** MNIST-superpixel graphs with their generated hard [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Ablation study results for Khan-GCL. Unsupervised learning results (in %) for graph classification on TU-datasets are [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Ablation study results regarding hyperparameters. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Ahmed, T.; and Sifat, M. H. R. 2024. GraphKAN: Graph Kolmogorov Arnold Network for Small Molecule-Protein Interaction Predictions. In ICML'24 Workshop ML for Life and Material Science: From Theory to Industry Applications

work page 2024

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arXiv 2025

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Chen, J.; Yuchi, X.; Yan, Z.; Dong, K.; and Li, H. 2025. KA - GAT : Kolmogorov Arnold based Graph Attention Networks

work page 2025

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2020

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work page 2022

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Chen, Y.; Frias, J.; and Gel, Y. R. 2024. TopoGCL: Topological Graph Contrastive Learning. arXiv preprint arXiv:2406.17251

work page Pith review arXiv 2024