Pith. sign in

REVIEW 3 major objections 5 minor 38 references

By drawing its grid points from a scrambled Sobol sequence, a Clifford-valued Kolmogorov-Arnold Network becomes an unbiased estimator of a convolution with variance O(n^{-1}), learning any such function with error that shrinks as the grid g

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 04:02 UTC pith:IQXNKVA2

load-bearing objection Useful empirical recipe with a flawed theoretical wrapper: the Sobol-grid expressivity lemma (IV.2) doesn't hold as stated, but the parameter-reduction observation is real enough to warrant serious review. the 3 major comments →

arxiv 2602.05977 v2 pith:IQXNKVA2 submitted 2026-02-05 cs.LG cs.AI

Clifford Kolmogorov-Arnold Networks

Matthias Wolff , Francesco Alesiani , Christof Duhme , Xiaoyi Jiang This is my paper
classification cs.LG cs.AI
keywords Kolmogorov-Arnold networksClifford algebrarandomized quasi-Monte CarloSobol sequencesradial basis functionshypercomplex neural networkscurse of dimensionality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces Clifford Kolmogorov-Arnold Networks (ClKAN), which move KAN-style learnable radial-basis activations from complex numbers into any Clifford algebra. To stop the number of grid points, and hence trainable weights, from growing exponentially with the algebra's dimension, the paper replaces the uniform lattice with a randomized quasi-Monte Carlo grid built from scrambled Sobol sequences. Its central claim is that on such a grid, the network's output Φ(x)=Σ w_g φ(x−g) is an unbiased estimator of the convolution h(x)=∫ g(y)φ(x−y)dy, with variance O(n^{-1}); therefore the model can learn any function that is a convolution of a smooth function with the radial kernel φ, using n grid points instead of 8^D. On complex-valued tasks ClKAN matches the CVKAN baseline, and on quaternion, Cl(2), Cl(1,1), and Cl(1,0,1) tasks the Sobol grid matches or outperforms the full grid while using a few percent of the parameters. The paper also proposes two RBF variants and three Clifford-aware batch-normalization schemes, with no single normalization strategy dominating in the experiments.

Core claim

The paper's key discovery is an identity between a KAN built on a scrambled Sobol grid and a randomized quasi-Monte Carlo integral estimate. Writing the network output as Φ(x)=Σ_{g∈G} w_g φ(x−g) and identifying the trained weights w_g with point evaluations of a function g(y) at the grid nodes, the authors prove that Φ(x) is an unbiased estimator of h(x)=∫_{[0,1]^d} g(y)φ(x−y)dy, with expected squared error bounded by Γσ²/n, i.e. O(n^{-1}). The immediate consequence is that the architecture can represent, with controlled error, every function that can be written as a convolution of a smooth g with the RBF kernel φ. Because the Sobol grid has a user-chosen number n of points, instead of the 8

What carries the argument

The Sobol grid—a scrambled (t,m,d)-net generated by a quasi-random Sobol sequence and scaled to a fixed range such as [−2,2]—is the object that carries the argument. It provides low-discrepancy grid points that cover the hypercube evenly in every realization, permitting a variance bound for scrambled nets. The central identity, Lemma IV.2, states that the RBF-sum Φ(x) over the Sobol grid forms an unbiased estimator of the convolution integral h(x), with variance O(n^{-1}); this lemma converts the network's expressivity into a numerical-integration statement. The two RBF variants (naive φ=exp(−∥x∥²) and Clifford φ=x exp(−∥x∥²)) determine whether the activation maps Cl→R or Cl→Cl, and the Clif

Load-bearing premise

The variance bound requires the trained weights w_g to be point evaluations of a fixed smooth function g(y) that is independent of the grid, but in the actual network the weights are free parameters optimized on the data, so the unbiasedness and O(n^{-1}) error need not hold for the trained model.

What would settle it

Train a Sobol-CliffordKAN on a function of the form h(x)=∫ g(y)φ(x−y)dy with a known smooth g, and record the empirical squared error as the number of grid points n ranges from, say, 8 to 2000. If the error does not decay like O(n^{-1}) or the estimator shows systematic bias—or if the fitted weights w_g at the grid points differ substantially from g evaluated at those points—then the convolution-estimator identification in Lemma IV.2 does not transfer to the trained network.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the convolution-estimator identity holds, a single ClKAN layer can approximate any convolution-kernel function in a Clifford algebra with error O(n^{-1}), roughly independent of the algebra dimension D.
  • On higher-dimensional algebras such as Cl(1,0,1), experiments show the Sobol grid reduces parameter count to about 2–6% of the full 8^D grid while matching or improving test error.
  • The kernel φ acts as a regularizing smoother: the network implicitly estimates the smoothed function g*φ, and in the β→∞, n→∞ limit the original g is recovered.
  • The three batch-normalization variants (node-wise, dimension-wise, component-wise) provide controls for keeping post-activation inputs inside the fixed grid range; without normalization training often collapses.
  • Re-sampling the Sobol grid for every cross-validation run yields an implicit ensemble effect, and an initial learning rate of 0.1 materially improves training stability in the reported experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A direct test of Lemma IV.2 would train Sobol-CliffordKAN on a known convolution h(x)=∫ g(y)φ(x−y)dy, measure the empirical squared error for increasing grid sizes n, and check whether the O(n^{-1}) rate and unbiasedness actually appear for the trained rather than the fixed function.
  • The transfer of the QMC variance bound to trained weights is the load-bearing assumption; a more defensible framing would treat the weights as a grid-dependent function g_n and analyze the realized estimator's bias-variance trade-off rather than fixed-function integration error.
  • Because the Clifford RBF φ(x)=x exp(−∥x∥²) is direction-preserving, the convolution interpretation suggests the architecture is specialized for signed, graded quantities; applying ClKAN to physics or robotics tasks with multivectors (for example Maxwell's equations or inverse kinematics) would test whether this geometric emphasis matters in practice.
  • The single-layer analysis could be extended compositionally: if each KAN layer is a convolution estimator, a deep stack composes or multiplies such smoothing operators; how the O(n^{-1}) error propagates through layers remains an open question worth testing.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces Clifford Kolmogorov-Arnold Networks (ClKAN), an extension of complex-valued KANs to arbitrary Clifford algebras. The main architectural contributions are two types of RBFs (naive and 'Clifford' RBF), three batch-normalization strategies for Clifford-valued activations, and a randomized quasi-Monte Carlo (RQMC) Sobol grid for generating the RBF centers. The central theoretical claim is Lemma IV.2, which states that the Sobol-grid network sum is an unbiased estimator of a convolution integral with O(n^{-1}) variance, and the paper uses this to argue that Sobol grids mitigate the exponential parameter growth of full grids. The experiments compare ClKAN with CVKAN baselines on complex-valued synthetic tasks, a holography dataset, and higher-dimensional Clifford algebras (e.g., Cl(2), Cl(0,2), Cl(1,1), Cl(1,0,1)).

Significance. If Lemma IV.2 were correct, the paper would provide a principled, scalable way to avoid the exponential growth of grid points in hypercomplex KANs, which is a relevant practical problem for scientific and engineering applications. The architecture extension itself is natural, and the paper covers several Clifford algebras and datasets; the commitment to release code is a strength. However, the theoretical support is invalid as written, and the empirical evidence is weakened by comparison-protocol mixing, best-configuration selection without error bars, and high run-to-run variance in several reported configurations. With a corrected theoretical statement (or an explicit demotion of the Sobol-grid claim to a heuristic) and more rigorous experimental reporting, the paper could become a useful contribution.

major comments (3)
  1. [IV-D, Lemma IV.2] The proof identifies the network sum Φ(x)=Σ_{g∈G} w_g φ(x−g) with the scrambled-net estimator ĥ_n(x)=(1/n)Σ_i g(y_i)φ(x−y_i) by setting g(y_i)=w_g. This identification fails twice. First, Φ contains no 1/n normalization; under the cited Property IV.1 the unscaled sum would have variance O(n), not O(n^{-1}). Second, Property IV.1 requires the integrand f_x(y)=g(y)φ(x−y) to be fixed before the sample is drawn; in a trained network the weights w_g are optimized on data and depend on the particular Sobol realization and the target function, so unbiasedness of Φ over random scramblings does not follow. The sentence 'training our KAN means training g(y) evaluated at grid points' is a reinterpretation, not a proof. The asserted O(n^{-1}) expressivity result is therefore unsupported.
  2. [V-A, Table I] The headline comparison mixes experimental protocols. The CVKAN baselines [7], [8] use learning rate 0.01 and report validation loss, while ClKAN uses learning rate 0.1, adds a separate test split, and reports test loss. Table I selects the best model across all RBF types, batch-normalization strategies, grid types, and architecture sizes, but gives no error bars, no number of configurations tried, and no significance testing. The claim that ClKAN 'achieves similar results' is therefore not established by this table; the improvement over [8] is confounded by the learning-rate increase.
  3. [VI-D, Table IV] The evidence for the higher-dimensional benefit of the Sobol grid is not robust. Several configurations in Table IV have standard deviations comparable to or exceeding the mean, e.g., square N_g=5: 0.102±0.199 and squaresquare N_g=8: 1.425±2.414. The conclusion that Sobol grids 'consistently' outperform the full grid is based on selected N_g values without multiple-comparison control. The reported instabilities undermine the parameter-reduction claim as a general statement; at minimum, per-fold results and a significance analysis are needed.
minor comments (5)
  1. [III-A and IV-D] The symbol n is used both for the vector-space dimension in Section III-A and for the grid size |G| in Section IV-D. The (t,m,d)-net definition notes this, but the ambiguity makes Lemma IV.2 and Eq. (5) hard to follow.
  2. [IV-A, Eq. (1b)] Calling x φ(x) a 'radial basis function' is misleading, since the term is not radial (it depends on direction). Suggest a name such as 'Clifford-weighted RBF'.
  3. [V-C and Tables II/IV] The phrase 'grid points per dimension' for the Sobol grid is ambiguous. For the full grid, N_g^D grid points are used; it is unclear whether the same total is used for the Sobol grid or whether N_g denotes the number of Sobol samples. The parameter counts suggest the latter for some tables, but this should be stated explicitly.
  4. [IV-D, Property IV.1] The proof of Property IV.1 cites [35] and [37] without specifying the exact theorem or the precise regularity conditions on f_x beyond L^{1+ε}. A self-contained statement of the theorem would help.
  5. [VI-B and VI-D, Figs. 4-5] The figures omit S-5 and S-7 'for better readability', but these are exactly the configurations that show high variance in Table IV. Please either include them or provide the omitted data in a supplementary table.

Circularity Check

0 steps flagged

No load-bearing circularity; central lemma uses external QMC theorems and has technical flaws, not circular reduction.

full rationale

The central theoretical claim (Lemma IV.2) is not circular: its variance bound is imported from the scrambled-net results of Owen [35,37], an external source, and the proof does not define the target result into its own assumptions. The identification Φ(x)=Σ_{g∈G} w_g φ(x−g)=ĥ_n(x) with g(y_i)=w_g is an attempted application of QMC theory, not a definitional equivalence of conclusion and premise; the proof's problems—the missing 1/n normalization, the treatment of data-dependent trained weights as fixed point evaluations of a smooth g, and the unverified L^{1+ε} condition—are technical invalidity of a theorem application, not circular reasoning. Self-citations to CVKAN [7] serve as architectural baseline and experimental comparison, not as support for the expressivity claim. The empirical parameter-reduction contribution is benchmarked against the full-grid CVKAN and is therefore externally falsifiable. The paper's own admission that some setups 'should work in theory' but show high variance (Section VI-E) is a stability/correctness concern, not a circularity indicator. No circular step meeting the evidentiary standard (definitional identity, fitted parameter renamed as prediction, or load-bearing self-citation) is present.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central theoretical claim rests on two unverified domain assumptions: that trained weights can be viewed as a fixed function sampled by the QMC net, and that the target class is convolutional with a smooth kernel. The empirical conclusions depend on hand-set hyperparameters (N_g, grid range, learning rate) that are tuned per dataset. No new physical entities are introduced.

free parameters (3)
  • Grid points per dimension N_g = 2–8 (per experiment)
    The number of Sobol grid points per dimension is chosen per dataset and is the key knob controlling parameter count; the paper reports the best N_g per dataset, so results depend on this selection.
  • Grid range = [−2, 2]
    Fixed grid range for each dimension, inherited from CVKAN; chosen by hand and not adapted, affects coverage of inputs.
  • Learning rate = 0.1
    Increased from 0.01 in the baseline to 0.1; the authors report this change alone markedly improves CVKAN performance, so the comparison to the lr=0.01 baseline is not controlled.
axioms (4)
  • standard math Kolmogorov-Arnold representation theorem (KAT) justifies single-variable edge functions.
    Used as motivation in Section I; not proved in this paper.
  • standard math Scrambled (t,m,d)-net variance bound: E[||ĥ_n − h||^2∞] ≤ Γ σ²/n (Owen & Rudolf).
    Invoked as Property IV.1; requires the integrand f_x(y) to be smooth and in L^{1+ε}, which is assumed without verification.
  • domain assumption The target function can be represented as a convolution ∫ g(y) φ(x−y) dy with a smooth g(y).
    Used in Lemma IV.2 to interpret the network as a QMC estimator; no argument that the function-fitting targets in Section V satisfy this representation.
  • domain assumption Clifford RBF basis and geometric-product weighting form a sufficiently expressive approximator.
    The architecture uses x φ(x) and weighted geometric products; no universality or approximation-rate theorem is given for ClKAN.

pith-pipeline@v1.3.0-alltime-deepseek · 12416 in / 16824 out tokens · 165522 ms · 2026-08-03T04:02:46.739144+00:00 · methodology

0 comments
read the original abstract

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford Algebra spaces. We propose the use of Randomized Quasi-Monte Carlo grid generation as a solution to the exponential scaling associated with higher-dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics-inspired tasks.

Figures

Figures reproduced from arXiv: 2602.05977 by Christof Duhme, Francesco Alesiani, Matthias Wolff, Xiaoyi Jiang.

Figure 1
Figure 1. Figure 1: Visualization of Cl(3) grades (scalars, vectors, bivec￾tors and trivectors). II. RELATED WORK A. Kolmogorov-Arnold Function Representation Kolmogorov-Arnold function representation theorem pro￾vides an alternative tool for function approximation [9], with the potential to address the curse of dimensionality [10]. The original KAN architecture has been extended to Convolu￾tional Neural Networks (CNNs) [11],… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of dimension-wise (orange), node-wise (green) and component-wise (blue) batch normalization [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of full grid, random grid, and quasi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overview of Mean Square Error (MSE) for all experiments for complex-valued synthetic function fitting tasks on [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Function-fitting experiments on higher dimensional Clifford algebras. Color represents Clifford algebra used, shape [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 6 linked inside Pith

  1. [1]

    A. N. Kolmogorov,On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables. American Mathematical Society, 1961

    1961

  2. [2]

    On the kolmogorov neural networks,

    A. Ismayilova and V . E. Ismailov, “On the kolmogorov neural networks,” Neural Networks, vol. 176, p. 106333, 2024

    2024

  3. [3]

    Space-filling curves and kolmogorov superposition-based neural networks,

    D. A. Sprecher and S. Draghici, “Space-filling curves and kolmogorov superposition-based neural networks,”Neural Networks, vol. 15, no. 1, pp. 57–67, 2002

    2002

  4. [4]

    KAN: Kolmogorov–arnold networks,

    Z. Liu, Y . Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljacic, T. Hou, and M. Tegmark, “KAN: Kolmogorov–arnold networks,” inICLR, 2025, pp. 70 367–70 413

    2025

  5. [5]

    A survey on kolmogorov-arnold network,

    S. Somvanshi, S. A. Javed, M. M. Islam, D. Pandit, and S. Das, “A survey on kolmogorov-arnold network,”ACM Computing Surveys, vol. 58, no. 2, pp. 1–35, 2025

    2025

  6. [6]

    KAN or MLP: A fairer comparison,

    R. Yu, W. Yu, and X. Wang, “KAN or MLP: A fairer comparison,” arXiv:2407.16674, 2024

    Pith/arXiv arXiv 2024

  7. [7]

    CVKAN: Complex-valued kolmogorov-arnold networks,

    M. Wolff, F. Eilers, and X. Jiang, “CVKAN: Complex-valued kolmogorov-arnold networks,” inIJCNN, 2025, pp. 1–9

    2025

  8. [8]

    Improved complex-valued Kolmogorov–Arnold networks with theoretical support,

    R. Che, L. af Klinteberg, and M. Aryapoor, “Improved complex-valued Kolmogorov–Arnold networks with theoretical support,” in24th EPIA Conference on Artificial Intelligence. Springer Nature Switzerland, 2026, pp. 439–451

    2026

  9. [9]

    The Kolmogorov superposition theorem can break the curse of dimensionality when approximating high dimensional functions,

    M.-J. Lai and Z. Shen, “The Kolmogorov superposition theorem can break the curse of dimensionality when approximating high dimensional functions,”arXiv:2112.09963, 2021

    arXiv 2021

  10. [10]

    How deep sparse networks avoid the curse of dimensionality: Efficiently computable functions are compositionally sparse,

    T. Poggio, “How deep sparse networks avoid the curse of dimensionality: Efficiently computable functions are compositionally sparse,”CBMM Memo, vol. 10, p. 2022, 2022

    2022

  11. [11]

    KANICE: kolmogorov-arnold networks with interactive convolutional elements,

    M. M. Ferdaus, M. Abdelguerfi, E. Ioup, D. Dobson, K. N. Niles, K. Pathak, and S. Sloan, “KANICE: kolmogorov-arnold networks with interactive convolutional elements,” in4th International Conference on AI-ML Systems (AIMLSystems), 2024, pp. 13:1–13:10

    2024

  12. [12]

    Convolu- tional kolmogorov-arnold networks,

    A. D. Bodner, A. S. Tepsich, J. N. Spolski, and S. Pourteau, “Convolu- tional kolmogorov-arnold networks,”arXiv:2406.13155, 2024

    Pith/arXiv arXiv 2024

  13. [13]

    Kolmogorov-arnold transformer,

    X. Yang and X. Wang, “Kolmogorov-arnold transformer,” inICLR, 2025

    2025

  14. [14]

    Exploring complex-valued convolu- tional Kolmogorov-Arnold networks for PolSaR image classification,

    Z. Kuang, H. Bi, Z. Lv, and C. Xu, “Exploring complex-valued convolu- tional Kolmogorov-Arnold networks for PolSaR image classification,” in IEEE International Geoscience and Remote Sensing Symposium, 2025, pp. 1945–1949

    2025

  15. [15]

    Incorporating arbitrary matrix group equivariance into KANs,

    L. Hu, Y . Wang, and Z. Lin, “Incorporating arbitrary matrix group equivariance into KANs,” inICML, 2025

    2025

  16. [16]

    Geometric Kolmogorov-Arnold Superposition Theorem,

    F. Alesiani, T. Maruyama, H. Christiansen, and V . Zaverkin, “Geometric Kolmogorov-Arnold Superposition Theorem,”arXiv:2502.16664, 2025

    Pith/arXiv arXiv 2025

  17. [17]

    Variational kolmogorov- arnold network,

    F. Alesiani, H. Christiansen, and F. Errica, “Variational kolmogorov- arnold network,”arXiv:2507.02466, 2025

    Pith/arXiv arXiv 2025

  18. [18]

    Kolmogorov- arnold networks meet science,

    Z. Liu, M. Tegmark, P. Ma, W. Matusik, and Y . Wang, “Kolmogorov- arnold networks meet science,”Phys. Rev. X, vol. 15, p. 041051, 2025

    2025

  19. [19]

    Residual kolmogorov-arnold network for enhanced deep learning,

    R. C. Yu, S. Wu, and J. Gui, “Residual kolmogorov-arnold network for enhanced deep learning,”arXiv:2410.05500, 2024

    arXiv 2024

  20. [20]

    Clifford algebra, geometric algebra, and applications,

    D. Lundholm and L. Svensson, “Clifford algebra, geometric algebra, and applications,”arXiv:0907.5356, 2009

    Pith/arXiv arXiv 2009

  21. [21]

    Clifford neural layers for pde modeling,

    J. Brandstetter, R. van den Berg, M. Welling, and J. K. Gupta, “Clifford neural layers for pde modeling,” inICLR, 2023

    2023

  22. [22]

    Geometric algebra transformers,

    J. Brehmer, P. De Haan, S. Behrends, and T. Cohen, “Geometric algebra transformers,” inNeurIPS, 2023

    2023

  23. [23]

    Clifford group equivariant neural networks,

    D. Ruhe, J. Brandstetter, and P. Forr ´e, “Clifford group equivariant neural networks,”NeurIPS, pp. 62 922–62 990, 2023

    2023

  24. [24]

    Geometric Clifford algebra networks,

    D. Ruhe, J. K. Gupta, S. De Keninck, M. Welling, and J. Brandstetter, “Geometric Clifford algebra networks,” inICML, 2023, pp. 29 306– 29 337

    2023

  25. [25]

    Con- formal geometric algebra for robotic vision,

    E. Bayro-Corrochano, L. Reyes-Lozano, and J. Zamora-Esquivel, “Con- formal geometric algebra for robotic vision,”Journal of Mathematical Imaging and Vision, vol. 24, pp. 55–81, 2006

    2006

  26. [26]

    Inverse kine- matics computation in computer graphics and robotics using conformal geometric algebra,

    D. Hildenbrand, J. Zamora, and E. Bayro-Corrochano, “Inverse kine- matics computation in computer graphics and robotics using conformal geometric algebra,”Advances in applied Clifford algebras, vol. 18, pp. 699–713, 2008

    2008

  27. [27]

    Geometric algebra: a computational framework for geometrical applications,

    L. Dorst and S. Mann, “Geometric algebra: a computational framework for geometrical applications,”IEEE Computer Graphics and Applica- tions, vol. 22, no. 3, pp. 24–31, 2002

    2002

  28. [28]

    Crumeyrolle,Orthogonal and symplectic Clifford algebras: Spinor structures

    A. Crumeyrolle,Orthogonal and symplectic Clifford algebras: Spinor structures. Springer Science & Business Media, 2013, vol. 57

    2013

  29. [29]

    Geometric algebra for electrical and electronic engineers,

    J. M. Chappell, S. P. Drake, C. L. Seidel, L. J. Gunn, A. Iqbal, A. Al- lison, and D. Abbott, “Geometric algebra for electrical and electronic engineers,”Proceedings of the IEEE, vol. 102, no. 9, pp. 1340–1363, 2014

    2014

  30. [30]

    Hestenes and A

    D. Hestenes and A. Lasenby,Space-time algebra. Springer, 2015

    2015

  31. [31]

    Automatic grid updates for Kolmogorov- Arnold networks using layer histograms,

    J. Moody and J. Usevitch, “Automatic grid updates for Kolmogorov- Arnold networks using layer histograms,”arXiv:2511.08570, 2025

    arXiv 2025

  32. [32]

    Free-knots Kolmogorov-Arnold network: On the analysis of spline knots and advancing stability,

    L. N. Zheng, W. E. Zhang, L. Yue, M. Xu, O. Maennel, and W. Chen, “Free-knots Kolmogorov-Arnold network: On the analysis of spline knots and advancing stability,”arXiv:2501.09283, 2025

    Pith/arXiv arXiv 2025

  33. [33]

    Random search for hyper-parameter opti- mization,

    J. Bergstra and Y . Bengio, “Random search for hyper-parameter opti- mization,”Journal of Machine Learning Research, vol. 13, no. 10, pp. 281–305, 2012

    2012

  34. [34]

    The distribution of points in a cube and the accurate evaluation of integrals,

    I. M. Sobol’, “The distribution of points in a cube and the accurate evaluation of integrals,”Zhurnal Vychislitel’noi Matematiki i Matem- aticheskoi Fiziki, vol. 7, no. 4, pp. 784–802, 1967

    1967

  35. [35]

    A strong law of large numbers for scrambled net integration,

    A. B. Owen and D. Rudolf, “A strong law of large numbers for scrambled net integration,”SIAM Review, vol. 63, no. 2, pp. 360–372, 2021

    2021

  36. [36]

    Scrambling Sobol’ and Niederreiter-Xing points,

    A. B. Owen, “Scrambling Sobol’ and Niederreiter-Xing points,”Journal of Complexity, vol. 14, no. 4, pp. 466–489, 1998

    1998

  37. [37]

    Randomly permuted (t, m, s)-nets and (t, s)-sequences,

    ——, “Randomly permuted (t, m, s)-nets and (t, s)-sequences,” in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Springer, 1995, pp. 299–317

    1995

  38. [38]

    Advancing mathematics by guiding human intuition with AI,

    A. Davies, P. Veli ˇckovi´c, L. Buesing, S. Blackwell, D. Zheng, N. Toma ˇsev, R. Tanburn, P. Battaglia, C. Blundell, A. Juh ´aszet al., “Advancing mathematics by guiding human intuition with AI,”Nature, vol. 600, no. 7887, pp. 70–74, 2021

    2021