Kolmogorov GAM Networks are all you need!

Sarah Polson; Vadim Sokolov

REVIEW 3 major objections 5 minor 61 references

Kolmogorov GAM Networks are all you need!

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a fixed fractal embedding followed by one shared outer function can represent any continuous function, making K-GAM a universal additive architecture.

desk verdict Unproven shared-outer-function claim and an internally inconsistent theorem statement sink the paper's central contribution; the honest negative Iris results don't rescue it. read the letter →

arxiv 2501.00704 v1 pith:534WCPRN submitted 2025-01-01 cs.LG stat.CO

Sarah Polson , Vadim Sokolov This is my paper · ORCID

classification cs.LGstat.CO MSC 41A6368T07

keywords KolmogorovSuperpositionTheoremGeneralizedAdditiveModelsKöppenfunctionK-GAMReLUnetworkstransformerkernelsmoothingembedding

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 introduces Kolmogorov GAM (K-GAM), a two-layer network in which the input is first mapped through a fixed fractal embedding built from the Köppen function and then through a single learned outer function $g$, implemented as a ReLU network, whose outputs are summed. The central claim is that this additive form is universal: any continuous function on the unit cube, and any finite dataset, can be represented this way, so the only trainable component is one univariate function. The authors argue that because the embedding is independent of the target function, K-GAM is a parameter-efficient alternative to transformers, which they reinterpret as kernel smoothers. They illustrate the construction on the Iris dataset and on a simulated five-dimensional regression benchmark, reporting that the architecture fits both tasks while using fewer parameters than standard deep models. The paper's own Iris experiment shows the K-GAM model with higher AIC and BIC and lower out-of-sample accuracy than a classical GAM, so the contribution is the representation theorem and architecture rather than an empirical win.

What carries the argument

The load-bearing mechanism is the pair consisting of the Köppen function $\psi$ and the shared outer ReLU network $g$. The Köppen function is a recursive, fractal, Hölder-continuous univariate map whose translates are summed inside each feature $z_k$, producing a topological embedding of $[0,1]^n$ into $\mathbb{R}^{2n+1}$ that is independent of the target function. The outer function $g$ is a finite-width ReLU network trained with SGD, and the theorem's power lies in this separation: all topological information is fixed in the embedding, so the learning problem becomes the estimation of a single univariate function.

What would settle it

Take a target that provably needs different outer functions, such as $f(x_1,x_2)=x_1x_2$, and compare the minimal $L^2$ approximation error of K-GAM with a single shared $g$ against the error of the full Sprecher-Köppen model with separate outer functions at matched total width; if the single-$g$ error stays bounded away from zero while the multi-$g$ error decays, the Theorem (K-GAM) is false.

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Extended reading notes

Core claim

The paper's central discovery, stated as the Theorem (K-GAM), is that any function and any dataset can be written as a generalized additive model whose features come from a fixed topological embedding rather than from learned weights: $$y_i = \sum_{k=1}^{2n+1} g(z_k), \qquad z_k = \sum_{j=1}^{n} \lambda_k \psi(x_j + \varepsilon_k) + k,$$ where $\psi$ is the Köppen function and $g$ is a single outer function shared by all $k$. The argument takes the Sprecher-Köppen refinement of Kolmogorov's superposition theorem and replaces each of the $2n+1$ outer functions $\phi_j$ with one ReLU network $g$, citing the ReLU approximation results of Montanelli et al. for the outer functions. The paper's interpretation is that this separates feature engineering from learning: the fractal embedding carries all topological information about the input space and is independent of $f$, while the trainable part reduces to a single univariate function.

Load-bearing premise

The universality claim rests on the assumption that the $2n+1$ distinct outer functions required by Kolmogorov's superposition theorem can be replaced by one shared ReLU network $g$ without losing expressive power; the paper asserts this replacement but does not prove it.

Editorial extensions

If this is right

If the Theorem (K-GAM) is correct, the parameter count of a predictor is set by the width of the single outer function $g$, not by the input dimension or dataset size.
Because the Köppen embedding is independent of the target function, the same embedding can be reused across learning tasks as a fixed dictionary or look-up table.
The sum over $2n+1$ parallel terms makes both training and inference parallelizable, so K-GAM is computationally attractive at scale.
Under the paper's reading of transformers as kernel smoothers, K-GAM supplies a lean additive alternative for sequence learning.
The small parameter count leaves room for Bayesian learning and posterior uncertainty quantification.

Reading between the lines

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

Editorial extension: If the single-$g$ collapse is valid, K-GAM would imply that expressivity comes from architecture rather than raw parameter scale, undercutting the scaling rationale for ever-larger transformer stacks.
Editorial extension: A natural stress test is to vary the smoothness or coordinate dependence of the target function; because the Köppen embedding is fixed and coordinate-blind, high-dimensional tasks with heterogeneous coordinate roles would reveal whether universality survives in practice.
Editorial extension: The non-differentiability of $\psi$ blocks gradient flow into the embedding, so a smoothed or learnable approximation of the Köppen function is a testable modification that would trade fidelity for trainability.
Editorial extension: The Iris experiment shows K-GAM trailing a classical GAM, so the practical case for K-GAM likely rests on high-dimensional or large-vocabulary tasks rather than small tabular data.

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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

Quick take: this is a well-intentioned but mathematically ungrounded attempt to recast Kolmogorov's Superposition Theorem as a GAM with a single outer function. The central theorem (Theorem K-GAM, Section 3) is not proven as stated, and the empirical results actually argue against the architecture.

What's genuinely new is the specific proposal to use the Köppen-Sprecher embedding as input-independent features for a GAM, plus the hope that a single shared outer g suffices. The paper does a decent job surveying KST and its recent ML connections (Montanelli-Yang, Schmidt-Hieber, KANs), and it honestly reports that K-GAM underperforms a classical GAM on Iris — higher AIC and BIC, lower accuracy. That transparency is to its credit.

The soft spots are load-bearing. The proof collapses 2n+1 distinct outer functions into one g with the line 'set g = φ_j for each j' and then 'replace each φ_j with a single ReLU network g.' That is not a derivation; it is an assertion. KST gives distinct outer functions, and a single continuous g can encode them only if the ranges of the inner features are pairwise disjoint, or some other special condition holds. The paper neither states nor proves such a condition. The theorem statement is also internally inconsistent: the z_k formula uses λ_k, which does not depend on the input coordinate j, while the proof uses coordinate-dependent λ_i. So even the displayed equation cannot be checked as printed.

The experiments are toy-scale (Iris, one simulated example), with no code or data, and the transformer comparison is purely conceptual — no transformer was run. The paper would need a correct proof of a single-g universality result, or at minimum a careful statement of conditions under which a shared outer function works, before it could be taken seriously. As is, it reads as a restatement of KST with a nice diagram and a cautionary data analysis.

Who is this for? Maybe someone wanting a quick survey of KST in an ML context, but not for the architectural claim. I would not cite the main theorem. A serious referee would have to engage with the gap, but most would recommend rejection, and I would agree.

Referee Report

3 major / 5 minor

Summary. The paper proposes Kolmogorov Generalized Additive Models (K-GAM), an architecture in which the input is first mapped through a fixed Köppen embedding ψ into 2d+1 one-dimensional features, and the output is a sum of evaluations of a trainable single-variable outer function g (implemented as a ReLU network) on those features. The central theoretical assertion is Theorem (K-GAM) in Section 3, which claims that any function and dataset can be represented in this form with a single shared outer function. The paper also discusses p-adic embeddings, Brillinger/PLS dimension reduction, kernel-smoothing interpretations of transformers, and illustrates the method on the Iris dataset and a synthetic five-dimensional regression problem.

Significance. If the single-g collapse were valid, K-GAM would provide a strikingly parameter-efficient universal additive architecture and a concrete bridge between KST, GAMs, and KANs. The literature review usefully collects classical KST constructions and draws attention to the Köppen embedding. However, the key theorem is not proved as written, and the only quantitative comparison (Iris) shows that K-GAM has higher AIC/BIC, lower R², lower out-of-sample accuracy, and higher RMSE than a classical GAM. The simulated-data section reports no quantitative results at all. Thus the advertised claims of universality, efficiency, and competitiveness with transformers are not established by the manuscript. The paper reads more as a research proposal or survey than as a validated architecture paper.

major comments (3)

[Section 3, Theorem (K-GAM) and proof] The proof of Theorem (K-GAM) does not establish the single-shared-g representation. The Sprecher-Köppen form quoted immediately before the theorem is f(x)=Σ_{j=0}^{2p} φ_j(Σ_i λ_i φ(x_i+ja)), with distinct outer functions φ_j. The proof writes 'set g = φ_j for each j' and then replaces every φ_j by one ReLU network g. These two steps are incompatible unless all φ_j coincide; a single continuous g can represent distinct φ_j only if the images z_j([0,1]^n) are pairwise disjoint, or are otherwise arranged so that g can be defined piecewise, and no such separation property is stated or proved. The displayed z_k in the theorem also does not match the derivation: it uses λ_k as the coefficient of ψ(x_j+ε_k) although λ_k is independent of the coordinate index j, whereas the proof uses coordinate-dependent λ_i; the quantities ε_k and '+k' are not defined, and 2p is not defined before use. Finally, applying the representation pointwise to m data points does not prove a 'dataset' version of the theorem, since the theorem concerns continuous functions on a cube, not finite-sample interpolation, and no generalization statement is derived. Because the single-g collapse is the defining property of the proposed K-GAM architecture, this gap is load-bearing.
[Section 5.2, Tables 1–3] The empirical results contradict the paper's headline claims. On the Iris benchmark, K-GAM has AIC 207.3 versus 43.0 for classical GAM, BIC 258.1 versus 59.3, R² 0.591 versus 0.790, RMSE 0.29 versus 0.22, and out-of-sample accuracy 34/45 versus 42/45. The text's concluding sentence for the example ('the inability of the GAM model to capture the complex relationships') is not supported by these numbers. Even if that sentence refers to the K-GAM model, the comparison shows the proposed architecture performing substantially worse while using a less interpretable feature representation. No confidence intervals, repeated-seed results, or significance tests are reported, so the comparison is also not statistically grounded.
[Section 5.1 and Section 3.2] The simulated-data experiment is described qualitatively but no quantitative results are given. The paper states that the first model uses multiple g_i functions and the second uses a single g with width increased from 16 to 200, but it does not report training or test error, parameter counts, runtime, convergence, or comparisons with any baseline. Consequently the central efficiency claim—'far fewer parameters than current deep learning algorithms'—is unsubstantiated. The inference discussion in Section 3.2 cites approximation rates for other architectures, such as O(M^{-1}) for Kolmogorov spline networks, but no analogous rate is derived for the proposed K-GAM architecture, so the parameter-efficiency claim is not supported by the theoretical material either.

minor comments (5)

[Section 5.2] The heading reads 'Iris Daa', and the text uses 'KST-GAN' in place of K-GAM; 'tanformation' and 'classibal' are typos that should be corrected.
[Section 3.3] The displayed relation Cov(g(Xβ),X)/Cov(g(Xβ),Xβ) = β Var(X)/Var(Xβ) is asserted without derivation and appears to require more than the preceding Stein lemma; it should be proved or removed.
[Section 6] There is an unresolved 'Figure??' cross-reference, and the phrase 'As Kolmogorov might have said' is informal for a research paper.
[Section 3, proof of Theorem (K-GAM)] The proof cites 'Montanelli et al. [2020]' while the bibliography contains Montanelli and Yang (2020) and Montanelli, Yang, and Du (2020); the intended source and the precise ReLU-approximation statement should be disambiguated.
[Section 4] The treatment of transformers is purely conceptual, and the claim that K-GAM is a natural alternative is never tested; a small-scale comparison would be needed to support that claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the K-GAM representation is imported from the external Kolmogorov-Sprecher-Köppen theorem and the outer function is genuinely fitted to data; the single-g collapse in Theorem (K-GAM) is an unproven logical step, not a circular reduction.

full rationale

No circular reduction is present in the derivation chain. The K-GAM model is defined by taking the external Kolmogorov-Sprecher-Köppen representation from Section 2 as the feature embedding, and the outer function is then estimated from data; nothing is fitted to a subset of data and announced as an independent prediction. The Iris and simulated-data sections are empirical comparisons against classical GAM and GLM, not derivations that recycle their own assumptions. The self-citations that appear (Bhadra et al. 2024; Polson et al. 2017) are contextual references for kernel embeddings and Bayesian deep learning; they are not load-bearing for the universality theorem. The main weakness is in the proof of Theorem (K-GAM) in Section 3: the sentence 'set g = φ_j for each j' followed by 'we can replace each φ_j with a single ReLU network g' is an unproven collapse. Unless the ranges of the z_j features are pairwise disjoint so that one shared g can be defined piecewise on each range, the single-g conclusion does not follow from KST. The displayed z_k formula is also internally inconsistent with the proof, since it uses a coordinate-independent λ_k while the proof uses coordinate-dependent λ_i. These are omitted justifications and correctness risks, not cases where the conclusion is equivalent to its inputs by construction. Accordingly, the circularity score is low even though the mathematical rigor of the central theorem is questionable.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The core of the paper rests on KST and ReLU approximation from the literature. The only ad hoc-to-paper assumption is the single shared outer function g, which is unproven and load-bearing. Hyperparameters such as network width and Köppen truncation level k are chosen by hand without sensitivity analysis.

free parameters (3)

Outer ReLU network width (K) = not stated; 16 or 200 wide in experiments
Chosen by hand; affects empirical performance and parameter count.
Köppen truncation level k = k=6 for Iris, k=3-5 for figures
Finite approximation ψ_k used in place of the infinite limit ψ; no error bound or sensitivity analysis.
γ base for p-adic expansion = 10 in Figure 1; not specified in experiments
Theoretically γ ≥ d+2; the concrete value used in the Iris and simulated runs is not reported, so the experiment is under-specified.

assumptions (4)

standard math Kolmogorov Superposition Theorem with Sprecher-Köppen inner functions
The central construction relies on KST as cited in Section 2 and Section 3; it is an external theorem, not proved in the paper.
standard math ReLU networks can approximate the outer functions φ_j
The proof invokes Montanelli et al. 2020 for replacing φ_j by a ReLU network to arbitrary precision.
ad hoc to paper One shared outer function g preserves representational power
The proof replaces φ_j with a single g without proof; this is the critical unproven assumption and it is not in KST.
domain assumption Function class and smoothness assumptions for rates
Citation of minimax rates presumes β-Hölder smoothness and bounded input; the paper does not verify these for the empirical data.

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Cite this review

Pith. "Pith review of Kolmogorov GAM Networks are all you need!." pith.science (2026). https://pith.science/paper/534WCPRN

@misc{pith2026250100704,
  author       = {Pith},
  title        = {Pith review of: Kolmogorov GAM Networks are all you need!},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/534WCPRN}},
  note         = {Machine review of arXiv:2501.00704}
}

read the original abstract

Kolmogorov GAM (K-GAM) networks are shown to be an efficient architecture for training and inference. They are an additive model with an embedding that is independent of the function of interest. They provide an alternative to the transformer architecture. They are the machine learning version of Kolmogorov's Superposition Theorem (KST) which provides an efficient representations of a multivariate function. Such representations have use in machine learning for encoding dictionaries (a.k.a. "look-up" tables). KST theory also provides a representation based on translates of the K\"oppen function. The goal of our paper is to interpret this representation in a machine learning context for applications in Artificial Intelligence (AI). Our architecture is equivalent to a topological embedding which is independent of the function together with an additive layer that uses a Generalized Additive Model (GAM). This provides a class of learning procedures with far fewer parameters than current deep learning algorithms. Implementation can be parallelizable which makes our algorithms computationally attractive. To illustrate our methodology, we use the Iris data from statistical learning. We also show that our additive model with non-linear embedding provides an alternative to transformer architectures which from a statistical viewpoint are kernel smoothers. Additive KAN models therefore provide a natural alternative to transformers. Finally, we conclude with directions for future research.

Figures

Figures reproduced from arXiv: 2501.00704 by the authors.

**Figure 2.** Scatter plot of the simulated dataset The goal is to predict the function y(x) based on the input x. The dataset is often used as a benchmark dataset for regression algorithms due to its diverse mix of relationships (linear, quadratic, nonlinear, Gaussian random noise) between the input features and the target function. The plot of µ (no noise) vs y (noise) is shown in [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** KST architecture for the simulated dataset [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Plot of the single outer function g for the simulated dataset 5.2 Iris Daa We apply the KST architecture to an iris dataset. The iris dataset is a classic dataset in machine learning and statistics. It contains 150 observations of iris flowers. Each observation contai…

**Figure 4.** Figure 4: Examples of outer functions g0, g2, g6, g8 for the simulated dataset three classes of iris flowers: setosa, versicolor, and virginica. The dataset is often used as a benchmark dataset for classification algorithms. The dataset has 5 variables, which include four charac…

**Figure 6.** Figure 6: Scatter plots of the iris dataset We calculate the mean µSL of the Sepal Length column and use binary variable yi = Sepal Lengthi > µSL as the output. We use the other three flower characteristics x1, x2, x3 as input variables. We used classical GAM model to fit the da…

**Figure 7.** Figure 7: Generalized additive model for the iris dataset versus KST. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

61 extracted references · 55 canonical work pages

[1]

Computation for the Kolmogorov Superposition Theorem

Jonas Actor. Computation for the Kolmogorov Superposition Theorem . MS Thesis , Rice, May 2018

work page 2018
[2]

Sergio Albeverio, Andrei Khrennikov, and Peter E. Kloeden. Memory retrieval as a p -adic dynamical system. Biosystems, 49 0 (2): 0 105--115, February 1999

work page 1999
[3]

Neural Machine Translation by Jointly Learning to Align and Translate , September 2014

Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural Machine Translation by Jointly Learning to Align and Translate , September 2014

work page 2014
[4]

Universal approximation bounds for superpositions of a sigmoidal function

Andrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information theory, 39 0 (3): 0 930--945, 1993

work page 1993
[5]

Adaptive Control Processes : A Guided Tour

Richard Bellman. Adaptive Control Processes : A Guided Tour . Princeton University Press, 1961. ISBN 978-0-691-07901-1

work page 1961
[6]

Polson, Vadim Sokolov, and Jianeng Xu

Anindya Bhadra, Jyotishka Datta, Nicholas G. Polson, Vadim Sokolov, and Jianeng Xu. Merging two cultures: Deep and statistical learning. WIREs Computational Statistics, 16 0 (2): 0 e1647, 2024

work page 2024
[7]

Wav- KAN : Wavelet Kolmogorov-Arnold Networks , May 2024

Zavareh Bozorgasl and Hao Chen. Wav- KAN : Wavelet Kolmogorov-Arnold Networks , May 2024

work page 2024
[8]

u rgen Braun. An Application of Kolmogorov 's Superposition Theorem to Function Reconstruction in Higher Dimensions . PhD thesis, Universit \

J \"u rgen Braun. An Application of Kolmogorov 's Superposition Theorem to Function Reconstruction in Higher Dimensions . PhD thesis, Universit \"a ts-und Landesbibliothek Bonn, 2009

work page 2009

Show all 61 references

[9]

Statistical Modeling : The Two Cultures (with comments and a rejoinder by the author)

Leo Breiman. Statistical Modeling : The Two Cultures (with comments and a rejoinder by the author). Statistical Science, 16 0 (3): 0 199--231, August 2001

2001
[10]

Brillinger

David R. Brillinger. A Generalized Linear Model With `` Gaussian '' Regressor Variables . In Peter Guttorp and David Brillinger, editors, Selected Works of David Brillinger , Selected Works in Probability and Statistics , pages 589--606. Springer, New York, NY, 2012. ISBN 978-...

2012
[11]

Sparse grids

Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica, 13: 0 147--269, May 2004

2004
[12]

Chua and L

L.O. Chua and L. Yang. Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems, 35 0 (10): 0 1257--1272, October 1988

1988
[13]

On Kolmogorov 's representation of functions of several variables by functions of one variable

Mark Coppejans. On Kolmogorov 's representation of functions of several variables by functions of one variable. Journal of Econometrics, 123 0 (1): 0 1--31, November 2004

2004
[14]

A note on computing with Kolmogorov Superpositions without iterations

Robert Demb and David Sprecher. A note on computing with Kolmogorov Superpositions without iterations. Neural Networks, 144: 0 438--442, December 2021

2021
[15]

On nonlinear functions of linear combinations

Persi Diaconis and Mehrdad Shahshahani. On nonlinear functions of linear combinations. SIAM Journal on Scientific and Statistical Computing, 5 0 (1): 0 175--191, 1984

1984
[16]

Sur une classe d' \'e quations fonctionnelles

Ivar Fredholm. Sur une classe d' \'e quations fonctionnelles. Acta Mathematica, 27 0 (none): 0 365--390, January 1903

1903
[17]

Buma L. Fridman. An improvement in the smoothness of the functions in AN Kolmogorov 's theorem on superpositions. In Doklady Akademii Nauk , volume 177, pages 1019--1022. Russian Academy of Sciences, 1967

1967
[18]

Projection pursuit regression

Jerome H Friedman and Werner Stuetzle. Projection pursuit regression. Journal of the American statistical Association, 76 0 (376): 0 817--823, 1981

1981
[19]

Gramacy and Herbert K

Robert B. Gramacy and Herbert K. H. Lee. Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103 0 (483): 0 1119--1130, 2008

2008
[20]

Counterpropagation networks

Robert Hecht-Nielsen . Counterpropagation networks. Applied optics, 26 0 (23): 0 4979--4984, 1987 a

1987
[21]

Kolmogorov's mapping neural network existence theorem

Robert Hecht-Nielsen . Kolmogorov's mapping neural network existence theorem. In Proceedings of the International Conference on Neural Networks , volume 3, pages 11--14. IEEE press New York, NY, USA, 1987 b

1987
[22]

Additive models in high dimensions, May 2002

Markus Hegland and Vladimir Pestov. Additive models in high dimensions, May 2002

2002
[23]

Computer Model Calibration Using High-Dimensional Output

Dave Higdon, James Gattiker, Brian Williams, and Maria Rightley. Computer Model Calibration Using High-Dimensional Output . Journal of the American Statistical Association, 103 0 (482): 0 570--583, June 2008

2008
[24]

Projection pursuit

Peter J Huber. Projection pursuit. The annals of Statistics, pages 435--475, 1985

1985
[25]

Igelnik and N

B. Igelnik and N. Parikh. Kolmogorov's spline network. IEEE Transactions on Neural Networks, 14 0 (4): 0 725--733, July 2003

2003
[26]

A three layer neural network can represent any multivariate function, January 2022

Vugar Ismailov. A three layer neural network can represent any multivariate function, January 2022

2022
[27]

On the Kolmogorov neural networks, October 2023

Aysu Ismayilova and Vugar Ismailov. On the Kolmogorov neural networks, October 2023

2023
[28]

Lee K. Jones. A simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training. Ann. Statist., 20 0 (1): 0 608--613, 03 1992. doi:10.1214/aos/1176348546

1992
[29]

Khavinson

S. Khavinson. Best Approximation by Linear Superpositions (Approximate Nomography), volume 159. American Mathematical Soc., 1997

1997
[30]

Risk bounds for high-dimensional ridge function combinations including neural networks

Jason M Klusowski and Andrew R Barron. Risk bounds for high-dimensional ridge function combinations including neural networks. arXiv preprint arXiv:1607.01434, 2016

2016 arXiv
[31]

On the representation of continuous functions of several variables as superpositions of functions of smaller number of variables

AN Kolmogorov. On the representation of continuous functions of several variables as superpositions of functions of smaller number of variables. In Soviet. Math . Dokl , volume 108, pages 179--182, 1956

1956
[32]

The curse of dimensionality

Mario K \"o ppen. The curse of dimensionality. In 5th Online World Conference on Soft Computing in Industrial Applications ( WSC5 ) , volume 1, pages 4--8, 2000

2000
[33]

On the Training of a Kolmogorov Network

Mario K \"o ppen. On the Training of a Kolmogorov Network . In Jos \'e R. Dorronsoro, editor, Artificial Neural Networks --- ICANN 2002 , Lecture Notes in Computer Science , pages 474--479, Berlin, Heidelberg, 2002. Springer. ISBN 978-3-540-46084-8

2002
[34]

The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions , November 2023

Ming-Jun Lai and Zhaiming Shen. The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions , November 2023

2023
[35]

The optimal rate for linear kb-splines and lkb-splines approximation of high dimensional continuous functions and its application

Ming-Jun Lai and Zhaiming Shen. The optimal rate for linear kb-splines and lkb-splines approximation of high dimensional continuous functions and its application. arXiv preprint arXiv:2401.03956, 2024

2024 arXiv
[36]

A Structured Self-attentive Sentence Embedding , March 2017

Zhouhan Lin, Minwei Feng, Cicero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen Zhou, and Yoshua Bengio. A Structured Self-attentive Sentence Embedding , March 2017

2017
[37]

KAN 2.0: Kolmogorov-Arnold Networks Meet Science , August 2024 a

Ziming Liu, Pingchuan Ma, Yixuan Wang, Wojciech Matusik, and Max Tegmark. KAN 2.0: Kolmogorov-Arnold Networks Meet Science , August 2024 a

2024
[38]

Hou, and Max Tegmark

Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Solja c i \'c , Thomas Y. Hou, and Max Tegmark. KAN : Kolmogorov-Arnold Networks , June 2024 b

2024
[39]

George G. Lorentz. The 13th problem of Hilbert . In Proceedings of Symposia in Pure Mathematics , volume 28, pages 419--430. American Mathematical Society, 1976

1976
[40]

Bayesian Approach to Global Optimization : Theory and Applications , volume 37 of Mathematics and Its Applications

Jonas Mockus. Bayesian Approach to Global Optimization : Theory and Applications , volume 37 of Mathematics and Its Applications . Springer Netherlands, Dordrecht, 1989. ISBN 978-94-010-6898-7 978-94-009-0909-0

1989
[41]

Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem, May 2020

Hadrien Montanelli and Haizhao Yang. Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem, May 2020

2020
[42]

Deep ReLU networks overcome the curse of dimensionality for bandlimited functions, November 2020

Hadrien Montanelli, Haizhao Yang, and Qiang Du. Deep ReLU networks overcome the curse of dimensionality for bandlimited functions, November 2020

2020
[43]

E. A. Nadaraya. On Estimating Regression . Theory of Probability & Its Applications, 9 0 (1): 0 141--142, January 1964

1964
[44]

Partial least squares estimator for single-index models

Prasad Naik and Chih-Ling Tsai. Partial least squares estimator for single-index models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62 0 (4): 0 763--771, 2000. doi:https://doi.org/10.1111/1467-9868.00262
[45]

Niyogi, F

P. Niyogi, F. Girosi, and T. Poggio. Incorporating prior information in machine learning by creating virtual examples. Proceedings of the IEEE, 86 0 (11): 0 2196--2209, November 1998

1998
[46]

On Estimation of a Probability Density Function and Mode

Emanuel Parzen. On Estimation of a Probability Density Function and Mode . The Annals of Mathematical Statistics, 33 0 (3): 0 1065--1076, September 1962

1962
[47]

A deep machine learning algorithm for construction of the Kolmogorov-Arnold representation, June 2020

Andrew Polar and Michael Poluektov. A deep machine learning algorithm for construction of the Kolmogorov-Arnold representation, June 2020

2020
[48]

Deep Learning : A Bayesian Perspective

Nicholas G Polson, Vadim Sokolov, et al. Deep Learning : A Bayesian Perspective . Bayesian Analysis, 12 0 (4): 0 1275--1304, 2017

2017
[49]

Eric A. F. Reinhardt, P. R. Dinesh, and Sergei Gleyzer. SineKAN : Kolmogorov-Arnold Networks Using Sinusoidal Activation Functions , July 2024

2024
[50]

Nonparametric regression using deep neural networks with ReLU activation function

Johannes Schmidt-Hieber . Nonparametric regression using deep neural networks with ReLU activation function. The Annals of Statistics, 48 0 (4): 0 1875--1897, August 2020

2020
[51]

The Kolmogorov -- Arnold representation theorem revisited

Johannes Schmidt-Hieber . The Kolmogorov -- Arnold representation theorem revisited. Neural Networks, 137: 0 119--126, May 2021

2021
[52]

Sprecher

David A. Sprecher. On the structure of continuous functions of several variables. Transactions of the American Mathematical Society, 115: 0 340--355, 1965

1965
[53]

An improvement in the superposition theorem of Kolmogorov

David A Sprecher. An improvement in the superposition theorem of Kolmogorov . Journal of Mathematical Analysis and Applications, 38 0 (1): 0 208--213, April 1972

1972
[54]

Charles M. Stein. Estimation of the Mean of a Multivariate Normal Distribution . The Annals of Statistics, 9 0 (6): 0 1135 -- 1151, 1981. doi:10.1214/aos/1176345632

1981
[55]

FC-KAN : Function Combinations in Kolmogorov-Arnold Networks , September 2024

Hoang-Thang Ta, Duy-Quy Thai, Abu Bakar Siddiqur Rahman, Grigori Sidorov, and Alexander Gelbukh. FC-KAN : Function Combinations in Kolmogorov-Arnold Networks , September 2024

2024
[56]

Transformer Dissection : A Unified Understanding of Transformer 's Attention via the Lens of Kernel , November 2019

Yao-Hung Hubert Tsai, Shaojie Bai, Makoto Yamada, Louis-Philippe Morency, and Ruslan Salakhutdinov. Transformer Dissection : A Unified Understanding of Transformer 's Attention via the Lens of Kernel , November 2019

2019
[57]

The Nature of Statistical Learning Theory

Vladimir Vapnik. The Nature of Statistical Learning Theory. Springer science & business media, 1999

1999
[58]

Gomez, Lukasz Kaiser, and Illia Polosukhin

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention Is All You Need , August 2023

2023
[59]

Geoffrey S. Watson. Smooth Regression Analysis . Sankhy \=a : The Indian Journal of Statistics, Series A (1961-2002) , 26 0 (4): 0 359--372, 1964

1961
[60]

Simon N. Wood. Generalized Additive Models : An Introduction with R , Second Edition . Chapman and Hall/CRC , New York, 2 edition, May 2017. ISBN 978-1-315-37027-9

2017
[61]

W. A. Z \'u \ n iga-Galindo , B. A. Zambrano-Luna , and Baboucarr Dibba. Hierarchical Neural Networks , p- Adic PDEs , and Applications to Image Processing . Journal of Nonlinear Mathematical Physics, 31 0 (1): 0 63, September 2024

2024

Pith tools

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

[1] [1]

Computation for the Kolmogorov Superposition Theorem

Jonas Actor. Computation for the Kolmogorov Superposition Theorem . MS Thesis , Rice, May 2018

work page 2018

[2] [2]

Sergio Albeverio, Andrei Khrennikov, and Peter E. Kloeden. Memory retrieval as a p -adic dynamical system. Biosystems, 49 0 (2): 0 105--115, February 1999

work page 1999

[3] [3]

Neural Machine Translation by Jointly Learning to Align and Translate , September 2014

Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural Machine Translation by Jointly Learning to Align and Translate , September 2014

work page 2014

[4] [4]

Universal approximation bounds for superpositions of a sigmoidal function

Andrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information theory, 39 0 (3): 0 930--945, 1993

work page 1993

[5] [5]

Adaptive Control Processes : A Guided Tour

Richard Bellman. Adaptive Control Processes : A Guided Tour . Princeton University Press, 1961. ISBN 978-0-691-07901-1

work page 1961

[6] [6]

Polson, Vadim Sokolov, and Jianeng Xu

Anindya Bhadra, Jyotishka Datta, Nicholas G. Polson, Vadim Sokolov, and Jianeng Xu. Merging two cultures: Deep and statistical learning. WIREs Computational Statistics, 16 0 (2): 0 e1647, 2024

work page 2024

[7] [7]

Wav- KAN : Wavelet Kolmogorov-Arnold Networks , May 2024

Zavareh Bozorgasl and Hao Chen. Wav- KAN : Wavelet Kolmogorov-Arnold Networks , May 2024

work page 2024

[8] [8]

u rgen Braun. An Application of Kolmogorov 's Superposition Theorem to Function Reconstruction in Higher Dimensions . PhD thesis, Universit \

J \"u rgen Braun. An Application of Kolmogorov 's Superposition Theorem to Function Reconstruction in Higher Dimensions . PhD thesis, Universit \"a ts-und Landesbibliothek Bonn, 2009

work page 2009