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REVIEW 4 major objections 5 minor 2 cited by

KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that KAN-based attention scoring has nearly unlimited ranking expressive power under bounded parameters and consistently outperforms linear and MLP scoring in attentive GNNs.

desk verdict Useful drop-in attention upgrade with a solid empirical story, but the theoretical expressiveness claims rest on a broken inequality and a contrived instance. read the letter →

arxiv 2501.13456 v4 pith:YQYMRKQW submitted 2025-01-23 cs.LG cs.AI

Taoran Fang , Tianhong Gao , Chunping Wang , Yihao Shang , Wei Chow , Lei Chen , Yang Yang This is my paper · ORCID
classification cs.LGcs.AI
keywords Kolmogorov-ArnoldattentionattentivegraphneuralnetworksscoringfunctionsmaximumrankingdistanceexpressivepowerB-splineKANnodeclassification
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 tries to establish that in attentive graph neural networks the scoring function—the part that decides how much each neighbor matters—is a real bottleneck, and that swapping it for a Kolmogorov-Arnold network (KAN) removes the bottleneck. It first unifies nearly all existing attentive-GNN scoring functions as a fixed alignment step followed by a learnable score mapping, then introduces Maximum Ranking Distance (MRD) to measure how far a scoring family can be from any desired neighbor ranking. Under bounded parameters, the paper shows linear and MLP scorers have positive MRD lower bounds, while a single-layer KAN with zero-order B-splines can push MRD below any positive threshold, giving it essentially arbitrary ranking power. The empirical claim is that KAA-enhanced models outperform their original backbones across all tested node- and graph-level tasks, with some improvements exceeding 20 percent.

What carries the argument

Central object: the unified scoring function $s(h_i,h_j) = \Psi \circ \mathrm{AF}(h_i,h_j)$, where $\mathrm{AF}$ is a fixed, non-learnable alignment (concatenation, absolute difference, or dot product) and $\Psi$ is the learnable score mapping. KAA sets $\Psi$ to a single-layer KAN whose univariate functions are sums of modified zero-order B-splines $\phi_k(x) = \sum_{l=1}^d c_{k,l} B^*_{k,l}(x)$. The load-bearing mechanism is the lookup-table effect: with grid size $d$, a row of the alignment matrix that contains exactly one coordinate divisible by $d$ activates exactly one spline coefficient, so the $d^2$ coefficients give one free parameter per row, enough to encode any target score assignment and hence any ranking of the $N=d^2$ nodes.

What would settle it

Collect the alignment vectors $\mathrm{AF}(h_i,h_j)$ from a trained backbone on a benchmark dataset and test whether each row contains exactly one coordinate divisible by the B-spline grid size $d$, as Appendix A.7 requires; if real rows are duplicate or lack this pattern, Proposition 3's bound does not apply to them. Independently, run KAA against each original scorer with matched parameter counts and the same hyperparameter search; the paper's universal empirical claim would be contradicted by any dataset on which a KAA-enhanced model does not beat its backbone.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the score mapping inside attentive-GNN attention—not the feature transformation—is the limiting component, and that replacing it with a Kolmogorov-Arnold network removes the limit. The paper proves for its constructed alignment matrix $P \in \mathbb{R}^{N\times d}$ that a linear scorer satisfies $\mathrm{MRD}(S_{\mathrm{LT}}, P) \ge \sqrt{\frac{1}{12}(N^3 - N - d^3 + 3d^2 - 2d)}$, that a two-layer MLP scorer has both upper and lower bounds of the same order, and that a single-layer KAN with modified zero-order B-splines satisfies $\mathrm{MRD}(S_{\mathrm{KAA}}, P) \le \delta$ for any $\delta > 0$, giving Theorem 1's ordering $\mathrm{MRD}(S_{\mathrm{KAA}}, P) \le \mathrm{MRD}(S_{\mathrm{MLP}}, P) \le \mathrm{MRD}(S_{\mathrm{LT}}, P)$. Empirically, KAA versions of GAT, GLCN, CFGAT, GT, and SAN outperform their originals on every evaluated node- and graph-level dataset, with average node-task gains around 1.6%, graph-task gains around 7.7%, and individual gains exceeding 20%.

Load-bearing premise

The theoretical advantage is proven for one specially structured set of node-pair inputs, namely a circulant alignment matrix in which each row activates exactly one spline coefficient, and the paper concedes that for other inputs, such as all-identical rows, every scoring function collapses to the same ranking.

Editorial extensions

If this is right

  • KAA can be applied to nearly any attentive GNN by replacing only the learnable score mapping, leaving the alignment function, aggregation, and training pipeline unchanged.
  • At matched or slightly lower parameter counts, KAA scorers have strictly smaller maximum ranking distance than linear and MLP scorers on the paper's construction, so the theoretical expressive-power ordering favors KAA.
  • KAA-enhanced GAT, GLCN, CFGAT, GT, and SAN beat their original versions on all evaluated datasets, with average improvements of about 1.6% on node-level tasks and 7.7% on graph-level tasks, and individual gains over 20%.
  • KAA also improves advanced attentive GNNs such as SuperGAT, HAT, NAGphormer, and SGFormer, so the benefit is not limited to the basic backbones.
  • The success locates the bottleneck in the score mapping rather than in the feature transformation, complementing earlier KAN-GNN work that used KANs to transform node features.

Reading between the lines

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

  • Editorial inference: the theory's lookup-table proof depends on the B-spline grid lining up with the alignment matrix; on real learned representations that lack this exact divisibility structure, the provable MRD advantage may shrink, which is an empirical question the paper does not resolve.
  • Editorial inference: if the unified scoring view is right, the same KAN replacement could be tested in other attention architectures beyond graphs, such as sequence Transformers, where the dot-product score is also a low-capacity linear mapping.
  • Editorial inference: a direct extension would be to measure, on real trained models, how close the matrix of alignment vectors is to the paper's circulant template, and to correlate that closeness with the size of the observed KAA gain.
  • Editorial inference: the MRD metric itself could serve as a diagnostic for scoring functions in retrieval and ranking tasks, where what matters is exactly the ordering induced by a learned score.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes Kolmogorov-Arnold Attention (KAA), a drop-in replacement for the scoring function in attentive GNNs, and introduces Maximum Ranking Distance (MRD) as a metric for the expressive power of scoring functions. The authors claim a theoretical hierarchy MRD(S_KAA) ≤ MRD(S_MLP) ≤ MRD(S_LT) under limited parameters, and support this with extensive node- and graph-level experiments on five backbone architectures showing consistent performance gains, some exceeding 20%.

Significance. If the theoretical hierarchy were established, the work would provide a principled explanation for why KAN-based scoring can outperform linear and MLP scoring in attentive GNNs, and the empirical study is a useful contribution in its own right. The experiments are detailed, cover multiple backbones and tasks, and include comparisons against GATv2 and other KAN-GNN variants. The code is released. However, the central theoretical claim is not proven as written: the proof of Proposition 1 contains an invalid max/min exchange, and Proposition 3's near-zero MRD is achieved by a lookup-table construction tailored to a specific alignment matrix, so the advertised 'nearly infinite expressive power' is an overstatement.

major comments (4)
  1. [Appendix A.5, Eqs. (32)–(34)] The proof of Proposition 1 moves the max over π′ inside the min over s′ and claims MRD ≥ min_{s′} sqrt(max_{π′} Σ (s′(P_i)−π′^{-1}(i))^2). For a max over one variable and a min over another, the valid inequality is max min ≤ min max, so the displayed direction is unjustified and can fail on simple finite examples. This invalidates the lower bound in Eq. (14) for linear-transformation attention. Since Theorem 1 relies on this bound to compare MRD(S_MLP) with MRD(S_LT), the proposed hierarchy is not established as written.
  2. [Appendix A.5, Eq. (35)–(38)] Even after the exchange, the evaluation of min_{a,b} max_{π′} Σ (a i + b − π′^{-1}(i))^2 is not carried out correctly: the authors first fix the 'uniformly distributed' worst-case permutation and then minimize over a and b, which computes something like max_{π′} min_{a,b} rather than the required min_{a,b} max_{π′}. The minimax value of this quantity is not derived, so the closed-form expression in Proposition 1 is unsupported regardless of the exchange issue.
  3. [Appendix A.7, Proposition 3 and Eq. (70)–(78)] The near-zero MRD for KAA is obtained by a construction where the alignment matrix P is the first d columns of an N×N circulant matrix with N=d^2 and the zero-order B-spline grid is chosen so that each row of P activates exactly one of the d^2 coefficients. This makes the KAN a lookup table with one learnable parameter per input row—i.e., the scoring function can memorize any ranking on this specific P. The paper itself concedes in Appendix A.4 that the composition of P controls the analysis (e.g., if P=1, all scoring functions are identical). The abstract's unqualified claim of 'nearly infinite expressive power' is therefore not supported by the theory; at most, the construction shows that a specially designed KAN can realize arbitrary rankings on a specially designed alignment matrix.
  4. [Theorem 1, Section 4.2.3] The ordering MRD(S_KAA, P) ≤ MRD(S_MLP, P) ≤ MRD(S_LT, P) depends on Proposition 1's lower bound, Proposition 2's bounds, and Proposition 3's upper bound. Since Proposition 1's lower bound is invalid for the reasons above, the ordering is not proven. A correct lower bound for linear-transformation attention might be smaller than the MLP upper bound, in which case the hierarchy could fail. The authors should either provide a valid proof of the bound or explicitly restrict the theoretical claims to the empirical finding that KAA improves performance in practice.
minor comments (5)
  1. [Propositions 1–3 and Appendix A.7] There is a recurring typo: 'matirx' should be 'matrix' in the statements of Propositions 1, 2, and 3.
  2. [Appendix A.6, Eq. (48)] The displayed formula has an extra parenthesis: 'a − π′^{-1}(i))2' should be '(a − π′^{-1}(i))^2'.
  3. [Table 3 and Section 5.3] The 'Avg Imp' row mixes classification accuracies (higher is better) with regression MAE values (lower is better). The reported positive average improvements for regression tasks should be clarified as decreases in MAE, otherwise the 'improvement' sign convention is inconsistent.
  4. [Section 4.2.3, paragraph before Proposition 3] The phrase 's(hi, hj) = Pd k=1' is missing proper typesetting; the sum should be written as s(hi, hj) = Σ_{k=1}^d φ_k(AF(hi, hj)_k).
  5. [Abstract and Section 1] The abstract states 'nearly infinite expressive power' without the qualifications introduced in Appendix A.4 and A.7 (the specific circulant alignment matrix and the lookup-table-style B-spline grid). Please add a caveat to the abstract or intro so the claim matches the proof conditions.

Circularity Check

1 steps flagged · score 6.0 of 10

KAA's claimed 'nearly infinite' expressive power is built into the co-designed circulant P and B-spline grid, making the headline theoretical result a lookup-table construction; the experiments are independent.

  1. self definitional [Appendix A.7, proof of Proposition 3; abstract's 'nearly infinite expressive power' claim]
    "In our assumption, we have N = d^2, and according to Formula 22, the elements in P take values from [1, N]. Therefore, the grid size of B∗(·) we used is also d... For any Pj, its d-dimensional values are composed of d consecutive integers, so there exists exactly one k ∈ [1, d] that satisfies d | Pj,k. Therefore, only one B∗k(·) is activated... s(Pj) = cd−β,α+1... for any permutationπ ∈ Π, we can find a group ofcd−β,α+1 = π′−1(αd + β + 1) such that MRD(SKAA, P) can be made arbitrarily small."

    The proof gives the KAN d^2 B-spline coefficients and then chooses P (first d columns of a circulant matrix with N=d^2) and grid size d so that row j=αd+β+1 activates exactly one coefficient, c_{d−β,α+1}. The bound MRD(S_KAA,P)≤δ is obtained by setting that coefficient equal to the target rank π'^{-1}(j); the inner minimization is then zero for every π'. This is a per-row lookup table: the 'expressive power' is one free parameter per ranked item, which is the input of the construction, not a derived property of KAN's compositional form. The paper itself concedes in Appendix A.4 that 'the composition of the P matrix has a significant impact on subsequent analysis...

full rationale

The empirical part of the paper is not circular: KAA variants are compared against original backbones on held-out test splits, with no fitted parameter relabeled as a prediction. The self-citations in the references (e.g., Fang et al. 2024, Sun et al. 2022/2024) are contextual and not load-bearing; the formal comparisons use Brody et al. (2021) and standard KAN references, not a self-citation chain. The circularity burden is confined to the theoretical expressive-power claim. Proposition 3's proof constructs a special alignment matrix P and a matching B-spline grid so that the KAN reduces to an arbitrary lookup table with one coefficient per row; the result MRD→0 is therefore equivalent by construction to giving one free parameter per data point. The abstract then states this instance-dependent construction as an unqualified property of KAA. A separate mathematical defect, the max/min exchange in Appendix A.5 (Eqs. 32–34), is a correctness problem rather than circularity and is not scored here; it further weakens the MRD hierarchy but does not change the circularity finding.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theoretical comparison rests on the co-designed pair (P, spline grid): P is the first d columns of an N by N circulant matrix with N = d squared, and the zero-order B-spline grid is sized so each of the d squared rows activates exactly one of the d squared coefficients. This makes the KAN exactly as expressive as a lookup table on this instance, while the competitors are width-constrained and evaluated on the same instance. The MLP and linear lower bounds additionally depend on an unproved maximin/minimax exchange and on standard facts about extreme rank assignments. The empirical section contributes standard hyperparameters rather than new free parameters, but the deployed near-linear KAN settings mean the experiments do not isolate the nonlinear mechanism the theory advertises.

free parameters (3)
  • Analysis instance P (first d columns of an N by N circulant matrix, N = d squared) = N = d squared
    The entire MRD comparison is computed on this single constructed matrix; the KAA zero-MRD result depends on each row containing exactly one multiple of d, aligned with the spline grid, and no result is proven for generic alignment matrices. The requirement that scores equal ranks (Appendix A.4, Eq. 23) ties the analysis to this instance.
  • Zero-order B-spline grid size d = d (grid matching the multiples-of-d structure of P)
    Proposition 3 requires the grid to partition the range into d cells of width d with inputs restricted to integers in the circulant pattern, making each row activate exactly one coefficient and turning KAA into a lookup table with one parameter per row.
  • Per-dataset KAN and backbone hyperparameters = grid size, spline order, layers in {1,2,4,8}, {1,2,3}, {2,3,4} plus layers, hidden dim, heads, dropout, learning rate…
    Selected by search over the stated ranges (Section 5.1); most runs settle on a single-layer KAN with grid size 1 and order 1, i.e., an effectively linear scorer, which the reported gains do not separate from the nonlinear expressiveness claim.
assumptions (5)
  • standard math Kolmogorov-Arnold representation theorem (smooth multivariate functions decompose into sums of univariate compositions)
    Invoked in Section 3 (Eq. 7) as the basis of the KAN architecture.
  • ad hoc to paper A maximin/minimax exchange: max over target rankings of min over scoring functions is at least min over scoring functions of max over rankings
    Appendix A.5 Eqs. 32 to 33; the standard minimax inequality goes the opposite direction, so the lower-bound proofs of Propositions 1 and 2 are not established as written.
  • standard math Worst-case target rankings assign the ranks of the first N+1-d nodes in alternating extreme order
    Used in the closed-form MRD calculations in Appendices A.5 and A.6 and attributed to Davis 1979 on circulant matrices; the claim is plausible but not proved in the paper.
  • ad hoc to paper Replacing the affine post-ReLU column vectors c_i with single-spike vectors z_i cannot increase the achievable ranking error (Appendix A.6)
    Stated without proof in the MLP lower bound; the inequality in Eq. 63 points in the direction that requires Z to be the harder family, which is asserted rather than demonstrated.
  • domain assumption The circulant first-d-columns matrix P is representative of realistic alignment inputs
    The analysis inputs are integer-valued circulant vectors while real experiments use learned continuous node representations; transfer from the constructed P to the general claim is assumed in the abstract, not proven.

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Pith. "Pith review of KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks." pith.science (2026). https://pith.science/paper/YQYMRKQW

@misc{pith2026250113456,
  author       = {Pith},
  title        = {Pith review of: KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQYMRKQW}},
  note         = {Machine review of arXiv:2501.13456}
}
read the original abstract

Graph neural networks (GNNs) with attention mechanisms, often referred to as attentive GNNs, have emerged as a prominent paradigm in advanced GNN models in recent years. However, our understanding of the critical process of scoring neighbor nodes remains limited, leading to the underperformance of many existing attentive GNNs. In this paper, we unify the scoring functions of current attentive GNNs and propose Kolmogorov-Arnold Attention (KAA), which integrates the Kolmogorov-Arnold Network (KAN) architecture into the scoring process. KAA enhances the performance of scoring functions across the board and can be applied to nearly all existing attentive GNNs. To compare the expressive power of KAA with other scoring functions, we introduce Maximum Ranking Distance (MRD) to quantitatively estimate their upper bounds in ranking errors for node importance. Our analysis reveals that, under limited parameters and constraints on width and depth, both linear transformation-based and MLP-based scoring functions exhibit finite expressive power. In contrast, our proposed KAA, even with a single-layer KAN parameterized by zero-order B-spline functions, demonstrates nearly infinite expressive power. Extensive experiments on both node-level and graph-level tasks using various backbone models show that KAA-enhanced scoring functions consistently outperform their original counterparts, achieving performance improvements of over 20% in some cases.

Figures

Figures reproduced from arXiv: 2501.13456 by the authors.

Figure 1
Figure 1. The alignment of our proposed KAA and other applications of KAN. (a) Symbiotic re [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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

  1. Improving Memory Efficiency for Training KANs via Meta Learning

    cs.LG 2025-06 conditional novelty 6.0 of 10

    MetaKANs generates each KAN activation function from a shared prompt-conditioned meta-learner, cutting trainable parameters toward MLP level while retaining comparable or better accuracy on tested benchmarks.

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

    cs.LG 2025-05 conditional novelty 6.0 of 10

    Khan-GCL combines KAN encoders with coefficient-based critical feature identification to generate hard negatives and reports state-of-the-art graph classification results.

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    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

    1944

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.