{"id":"5741018d-28b0-4708-b4a1-9fdaedd2806d","arxiv_id":"2505.15103","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Khan-GCL combines KAN encoders with coefficient-based critical feature identification to generate hard negatives and reports state-of-the-art graph classification results.","lead":"The paper introduces a graph contrastive learning method that uses Kolmogorov-Arnold Networks as encoders and generates hard negative samples by perturbing the most informative feature dimensions identified from the network's B-spline coefficients. If the reported gains hold, it provides a new way to improve self-supervised graph representation learning without labels.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2's variance bound is false: dropping the off-diagonal terms in Eq. 26 is invalid because those terms can be positive, so the discriminative-score justification is unsupported.","rationale":"The paper's central claim is empirical SOTA for KAN-based GCL with CKFI hard negatives. The weakest load-bearing link is the theoretical justification of the discriminative dimension score rho_j in Section 4.2. Proposition 2's bound is used to argue that high coefficient variance marks high functional variance. Appendix A.2's derivation of Eq. 27 drops the off-diagonal terms 2M(1) sum d_i d_{i+1} + 2M(2) sum d_i d_{i+2} + 2M(3) sum d_i d_{i+3} after noting M(k) >= 0. But nonnegativity of M(k) does not make these terms nonnegative; the signs depend on the centered coefficient deviations d. A low-frequency cosine vector has zero mean, so it is admissible, and for large n its Rayleigh quotient approaches the largest eigenvalue of the B-spline Gram matrix, which strictly exceeds M(0) because the off-diagonal bands are positive. Thus the inequality is false and the formal link between coefficient variance and output variance is broken. This matters because Eq. (9) perturbs dimensions in proportion to rho_j and delta_j; if large rho_j does not imply large functional variance, the 'discriminative' hard negatives lose their stated mechanism. The empirical ablation (Figure 4) shows targeted perturbations outperform random ones, so the method may still work as a heuristic, but the proof as written is incorrect and should either be fixed or removed. The lack of code and undisclosed HOSVD ranks are additional reproducibility concerns, but they are secondary to the mathematical flaw. The verdict should remain conditional: the empirical claim is plausible but not fully supported until the proposition is corrected or its role is reframed, and code is provided.","tokens_in":19238,"tokens_out":5656,"duration_ms":51717,"concrete_test":"Construct the n x n Gram matrix M for uniform cubic B-splines with entries M_{ij} = integral B_i B_j, set d_i = cos(2 pi i / n) for n >= 8, and compute the ratio R = (d^T M d) / (M(0) ||d||^2). If R > 1, Proposition 2 is false. Additionally, compute the largest eigenvalue of M restricted to the zero-mean subspace d^T 1 = 0 and compare it with M(0); if it exceeds M(0), the bound fails for admissible coefficient deviations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2 (Section 4.2) claims Var[phi(x)] <= M(0) * sigma_c^2 for a uniform B-spline. The proof in Appendix A.2 expands Var[phi(x)] = d^T M d, where d_i = c_i - mean(c) and M_{ij} = integral B_i B_j, then asserts that since all M(k) >= 0, the off-diagonal terms 2M(1) sum d_i d_{i+1} + 2M(2) sum d_i d_{i+2} + 2M(3) sum d_i d_{i+3} can be dropped to obtain an upper bound. This step is invalid: nonnegativity of the entries M(k) does not make those quadratic terms nonnegative; their signs depend on the centered deviations d. For an admissible zero-mean vector such as d_i = cos(2 pi i / n), the quadratic form approaches lambda_max(M) ||d||^2 for large n, and because M has positive off-diagonal bands, lambda_max(M) > M(0). Thus the inequality is false. Equation (8) uses rho_j, an average of coefficient variances, as a discriminative score, and Equation (9) perturbs dimensions in proportion to rho_j and delta_j. Without a valid bound, the formal link between high coefficient variance and high functional variance is broken, and the stated theoretical justification for the discriminative component of CKFI is unsupported. The empirical SOTA claim may still hold as a heuristic, but this is a genuine correctness flaw in a named contribution.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19668,"tokens_out":6679,"duration_ms":59095,"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":[{"comment":"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":"Section 4.2, Proposition 2 and Appendix A.2 (Eqs. 26–27)"},{"comment":"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":"Section 4.2–4.3"},{"comment":"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.","section":"Section 5, Tables 1–3"}],"minor_comments":[{"comment":"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.","section":"Appendix C, Algorithm 2"},{"comment":"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.","section":"Appendix E.1"},{"comment":"The column header 'A VG' appears to be a typo for 'Avg'.","section":"Tables 1 and 2"},{"comment":"The phrase 'balance between expensiveness and risks' should read 'balance between expressiveness and risks'.","section":"Section 1, Introduction"},{"comment":"The heading contains the typo 'Algorothm' and should read 'Algorithm'.","section":"Appendix B"},{"comment":"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.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The main risk for the editor is that the paper's headline claim depends on two items that are currently not in publishable shape: the false variance bound and the absence of a common-baseline comparison. I would not reject on the empirical evidence alone, because the ablations are internally consistent and the method is plausible, but the theoretical flaw must be fixed (or the theory explicitly downgraded to a heuristic) before acceptance. The authors should also be asked to release code, since the HOSVD-based independent score has several implementation choices (truncation ranks, least-squares projection) that are hard to reproduce from the text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nWhere 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.\n\nTwo 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.\n\nBottom 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.","headline":"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.","tokens_in":20102,"tokens_out":3172,"would_cite":false,"duration_ms":28699,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["graph contrastive learning","Kolmogorov-Arnold networks","hard negatives","B-spline coefficients","critical feature identification","self-supervised representation learning","graph classification","tensor decomposition"],"falsifier":"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.","tokens_in":19072,"feed_emoji":"🔗","tokens_out":15243,"duration_ms":120247,"temperature":0.7,"pith_summary":"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.","feed_headline":"Graph learner with KAN encoder tops benchmarks via hard negatives","feed_subtitle":"Replacing MLP encoders with KANs and perturbing critical dimensions yields sharper graph representations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Kolmogorov-Arnold Network with B-spline parameterized activations; the KAN encoder and its coefficient tensor are the basis of Khan-GCL.","marker":"(Liu et al. 2024)"},{"why":"GraphCL supplies the contrastive loss, augmentation scheme, and MLP backbone sizes that Khan-GCL replaces with KAN, and serves as a primary baseline.","marker":"(You et al. 2020)"},{"why":"Provides the NT-Xent-style contrastive loss that Equation (1) of the paper uses to train positive and negative pairs.","marker":"(Chen et al. 2020)"},{"why":"Establishes the hard-negative criteria and shows that merely enlarging batch size can degrade performance, motivating targeted hard-negative construction.","marker":"(Kalantidis et al. 2020)"},{"why":"Hard-negative mining baseline for GCL that Khan-GCL's perturbation-based negatives are designed to improve upon.","marker":"(Xia et al. 2021)"},{"why":"Provides the higher-order singular value decomposition used to score the independence of each output dimension via reconstruction error.","marker":"(Tucker 1966; De Lathauwer, De Moor, and Vandewalle 2000)"},{"why":"Supplies the unsupervised evaluation protocol (SVM on pre-trained representations) and InfoGraph baseline used in the TU-dataset experiments.","marker":"(Sun et al. 2019)"},{"why":"GraphACL is the adversarial hard-negative baseline in the main comparison; the paper argues such adversarial negatives do not explicitly target critical semantic dimensions.","marker":"(Luo et al. 2023)"},{"why":"Provides the eight TU-datasets used for unsupervised graph classification evaluation.","marker":"(Morris et al. 2020)"},{"why":"Provides the Zinc-2M dataset used for large-scale pre-training in the transfer learning experiments.","marker":"(Sterling and Irwin 2015)"}],"fun_headline_variants":["KAN boosts graph contrastive learning with smarter hard negatives","Graph learning gets KAN-powered hard negatives for sharper features","Khan-GCL: KAN encoder and coefficient-driven hard negatives","Hard negatives from KAN coefficients set new graph learning SOTA"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["KAN boosts graph contrastive learning with smarter hard negatives","Graph learning gets KAN-powered hard negatives for sharper features","Khan-GCL: KAN encoder and coefficient-driven hard negatives","Hard negatives from KAN coefficients set new graph learning SOTA"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1314,"prompt_tokens":926,"completion_tokens":388,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":318}},"tokens_in":542,"tokens_out":388,"duration_ms":3561,"temperature":1.0,"reasoning_tokens":318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:25:03.708014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"B.; Pion, N.; Weinzaepfel, P.; and Larlus, D","cited_arxiv_id":null,"evidence_quote":"Establishes the hard-negative criteria and shows that merely enlarging batch size can degrade performance, motivating targeted hard-negative construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"GraphACL is the adversarial hard-negative baseline in the main comparison; the paper argues such adversarial negatives do not explicitly target critical semantic dimensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Zinc-2M dataset used for large-scale pre-training in the transfer learning experiments."}],"review_version":1}