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REVIEW 3 major objections 5 minor 1 cited by

KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forward

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read KANs claim to break the curse of dimensionality

desk verdict A readable KAN primer whose central dimension-free claim does not follow from the cited theorem. read the letter →

arxiv 2411.10622 v1 pith:R43LVNYB submitted 2024-11-15 cs.LG cs.NEstat.ML

Divesh Basina , Joseph Raj Vishal , Aarya Choudhary , Bharatesh Chakravarthi This is my paper · ORCID
classification cs.LGcs.NEstat.ML MSC 68T0741A1541A30
keywords Kolmogorov-ArnoldNetworksrepresentationtheoremcurseofdimensionalityB-splinesuniversalapproximationerrorscalingneuralnetworkarchitecture
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

This review argues that Kolmogorov-Arnold Networks (KANs), a neural architecture introduced in 2024, can escape the curse of dimensionality that limits traditional multilayer perceptrons. The paper explains the underlying mathematics—interpolation, B-splines, the universal approximation theorem, and the Kolmogorov-Arnold representation theorem—and presents the error bound that supports the claim. If correct, KANs would train accurate models on high-dimensional data with far fewer parameters and less data than MLPs. The review also stresses that the guarantee depends on the target function possessing a polynomial-like structure with smooth univariate components, a condition it does not verify for real-world tasks.

What carries the argument

The central object is the Kolmogorov-Arnold representation theorem (KAT), which states that any continuous multivariate function on [0,1]^n can be written as a sum of outer univariate functions composed with sums of inner univariate functions: f(x) = Σ_{q=1}^{2n+1} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)). KANs turn this into a neural architecture by replacing fixed activations with learnable univariate functions approximated by B-splines. The theorem from Liu et al. is the load-bearing error bound: with B-spline order k and grid size G, the C^m error decays as C $G^{{-k-1+m}}$, independent of dimension, which is the source of the claimed curse-of-dimensionality freedom.

What would settle it

Construct a continuous function on [0,1]^n that is known not to admit a smooth KAT representation—for example, a function whose KAT components are fractal or discontinuous, such as the maximum of n variables—and train a KAN on it for increasing n. If the empirical error grows with n in a way that violates C $G^{{-k-1+m}}$ (e.g., requires exponentially more grid points), the claimed dimension-free bound fails.

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Extended reading notes

Core claim

The paper's central claim is that KANs achieve true freedom from the curse of dimensionality. This rests on a theorem from Liu et al. stating that for any function admitting a KAN representation with (k+1)-times continuously differentiable B-spline components, the approximation error satisfies ||f - (Φ_{L-1}∘...∘Φ_0)x||_{C^m} ≤ C $G^{{-k-1+m}}$, where G is the grid size and C is a constant independent of the input dimension n. Because the error bound does not involve n, the review concludes that KANs can handle high-dimensional inputs without the exponential growth in data requirements that plagues standard neural networks. It further claims that KANs need fewer parameters and achieve better scaling exponents than MLPs, positioning them as a practical alternative for high-dimensional learning.

Load-bearing premise

The entire curse-of-dimensionality argument depends on the target function admitting a KAN representation with smooth B-spline components, and the review also assumes the underlying process has a higher-order polynomial expression; if the data contain discontinuities or lack such structure, the error bound and the freedom-from-curse conclusion collapse.

Editorial extensions

If this is right

  • KANs could train on high-dimensional inputs without the exponential data growth that MLPs require, because the approximation error does not scale with input dimension.
  • KANs may need far fewer parameters than MLPs to reach comparable accuracy, since each univariate function is directly parameterized by B-spline coefficients.
  • The error bound suggests that increasing the B-spline grid size G is a controlled way to improve accuracy, unlike the unbounded neuron count N(ϵ) in the universal approximation theorem.
  • KANs are best suited to problems where the target function is believed to have a smooth, polynomial-like structure; where that structure is absent, the guarantees do not apply.

Reading between the lines

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

  • The curse-of-dimensionality freedom is conditional: for a given real-world dataset, one must first verify that the true mapping lies in the KAN function class. Without that verification, the error bound offers no practical guarantee, and empirical performance could degrade with dimension.
  • A testable extension would be to compare KANs and MLPs on high-dimensional benchmark functions where the KAN representation condition is known to fail (e.g., functions with discontinuous or fractal KAT components) to see whether the claimed advantage disappears.
  • The review's emphasis on polynomial structure suggests KANs may be especially useful in scientific computing and symbolic regression, where such structure is often assumed; the paper does not explore this connection explicitly.
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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

3 major / 5 minor

Summary. The paper is a broad survey of Kolmogorov-Arnold Networks (KANs). It reviews interpolation, B-splines, perceptrons, the universal approximation theorem, the Kolmogorov-Arnold representation theorem, and the KAN architecture, and it argues from Theorem 1 of Liu et al. that KANs achieve 'true freedom from the curse of dimensionality.' It also claims that KANs require far fewer parameters and exhibit better scaling than MLPs. The manuscript contains no new derivations or experiments; its conclusions are drawn from cited work and from the authors' interpretation of the error bound.

Significance. If the advertised claim were valid, it would be a major result for high-dimensional learning, and the paper provides a reasonably accessible introduction to the mathematical ingredients of KANs. The pedagogical sections on splines and interpolation are clear, and the paper explicitly identifies the polynomial-structure assumption in Section 3, which is a useful caution. The paper also cites a broad set of recent KAN application papers. However, the central quantitative claim is not established: the quoted error bound does not control the dependence of the constant and grid size on the input dimension, and the paper leaves the unrelated-input question in Section 6 unanswered before repeating the strong conclusion in Section 7. The paper therefore overstates what the cited theorem supports.

major comments (3)
  1. [Abstract; Section 6, Theorem 1] The claim that Theorem 1 shows 'true freedom from the curse of dimensionality' is not supported by the stated bound. The bound ||f − (Φ_{L−1}∘...∘Φ_0)x||_{C^m} ≤ C G^{−k−1+m} only says that for a fixed f whose KAN representation has (k+1)-times continuously differentiable components, the error decreases as the spline grid G grows. It does not state that the constant C or the required G are independent of the input dimension n. A concrete example shows why this matters: f_n(x)=exp(x_1+...+x_n) is exactly representable by a KAN with Φ(t)=e^t on [0,n], but the B-spline error of approximating e^t on an interval of length n scales like e^n/G^{k+1} up to polynomial factors in n, so holding the error below ε forces G to grow exponentially in n. Thus the theorem, as quoted, is compatible with an exponential parameter/data requirement, and the Abstract's unconditional claim is not a consequence of it.
  2. [Sections 6 and 7] Section 6 raises the exact objection that would limit the curse-of-dimensionality claim: 'what happens when a model is trained with unrelated input and output features?' and asks whether the bound ε ≤ CG^{−k−1+m} still holds. No answer is given, and Section 7 then reasserts that KANs are promising for high-dimensional real-world tasks. This is load-bearing because Theorem 1 is conditional on f admitting a smooth KAN representation; an arbitrary mapping from unrelated features to labels need not satisfy that condition, so the error bound cannot be invoked for such tasks. The authors must either answer the question with a rigorous statement of the class of functions for which the bound holds, or explicitly restrict the conclusion to that class.
  3. [Sections 3 and 7] The paper's generalization from the theorem to practice relies on the assumption, introduced in Section 3, that 'the process we are trying to model has a higher-order polynomial expression.' Section 7 acknowledges that this structure 'can be challenging to prove' and proposes that it is 'reasonable to consider' whenever a polynomial form cannot be ruled out. That is not a verification: for image, video, or arbitrary regression tasks, there is no evidence that a KAN representation with smooth B-spline components exists. Without such evidence, the error bound does not apply and the claimed scalability advantages are not established. The conditional nature of the claim should be stated in the Abstract and Conclusion.
minor comments (5)
  1. [Equation (21)] Equation (21) states f : [0,1] → R, but the theorem applies to multivariate f : [0,1]^n → R; this is a typo that should be corrected.
  2. [Section 3.3] There are several typos, including 'the the rectified linear unit' in Section 3.3 and inconsistent capitalization 'KANS' in Section 6.
  3. [Figure 5 caption] The caption of Figure 5 says the B-spline interpolant manages to interpolate 'without needing any data,' which is misleading; the interpolant is constructed from the same 50 interpolation points used by the other methods.
  4. [References] The reference list relies heavily on Wikipedia entries for mathematical background (e.g., [16], [23]–[26], [29], [30]); for a formal review, primary sources would be more appropriate.
  5. [Section 2] The description of the Tschirnhaus transformation as reducing to 'n − 4 variables when n ≥ 7' is imprecise and is not used later in the paper; it should be rephrased or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the main KAN claims are conditional on an externally stated representation theorem, with any overreach being a correctness concern rather than circularity.

full rationale

This review makes no fitted predictions and performs no derivation that reduces to its own inputs. The central claim that KANs are not subject to the curse of dimensionality is explicitly imported from Theorem 1 of Liu et al., an approximation bound for functions that admit a smooth KAN representation. The paper states the theorem's hypothesis verbatim ("Suppose that a function f(x) admits a representation f(x) = (Phi_(L-1) composed with ... Phi_0)x ... where each Phi_(l,i,j) are (k+1)-times continuously differentiable") and its conclusion is a concrete grid-size bound. The premise and conclusion are not identical: the premise is representability by smooth univariate compositions, and the conclusion is a B-spline approximation rate in G. The paper may overstate the theorem's scope by inferring dimension-independence from a bound whose constant C depends on f, but that is a correctness or support-level concern, not circularity. The paper also flags the representability assumption itself in Section 7 ("it can be challenging to prove this structure in any given system, it is often easier to disprove by identifying discontinuities"), acknowledging the conditional nature of the claim. No parameters are fitted, no known result is renamed, and no self-citation is load-bearing. Therefore no circular steps are present.

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

The review introduces no fitted parameters and no invented entities. It depends on four external assumptions: the KAT theorem, UAT, the polynomial-structure assumption, and Theorem 1 from Liu et al. The first two are standard; the last two are load-bearing and unverified within the paper.

assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem holds for all continuous multivariate functions on [0,1]^n with continuous univariate functions phi_{q,p} and continuous outer functions Phi_q.
    Invoked in Section 3 and Section 4.2 as the foundation for KANs; accepted as a theorem.
  • standard math Universal approximation theorem holds and provides approximation without specifying construction.
    Used in Section 4.1 to contrast with KAT.
  • domain assumption The modeled process has a higher-order polynomial structure, making KAT-based decomposition applicable.
    Stated in Section 3: 'under the assumption that the process we are trying to model has a higher-order polynomial expression'; used in Section 7 to justify wide applicability.
  • domain assumption Theorem 1 from Liu et al.: approximation error for KANs with k-th order B-splines on grid G is O(G^{-k-1+m}) independent of input dimension.
    Quoted in Section 6 from reference [1] without proof; the review's curse-of-dimensionality claim rests entirely on this theorem.

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Cite this review

Pith. "Pith review of KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forward." pith.science (2026). https://pith.science/paper/R43LVNYB

@misc{pith2026241110622,
  author       = {Pith},
  title        = {Pith review of: KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forward},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R43LVNYB}},
  note         = {Machine review of arXiv:2411.10622}
}
read the original abstract

The curse of dimensionality poses a significant challenge to modern multilayer perceptron-based architectures, often causing performance stagnation and scalability issues. Addressing this limitation typically requires vast amounts of data. In contrast, Kolmogorov-Arnold Networks have gained attention in the machine learning community for their bold claim of being unaffected by the curse of dimensionality. This paper explores the Kolmogorov-Arnold representation theorem and the mathematical principles underlying Kolmogorov-Arnold Networks, which enable their scalability and high performance in high-dimensional spaces. We begin with an introduction to foundational concepts necessary to understand Kolmogorov-Arnold Networks, including interpolation methods and Basis-splines, which form their mathematical backbone. This is followed by an overview of perceptron architectures and the Universal approximation theorem, a key principle guiding modern machine learning. This is followed by an overview of the Kolmogorov-Arnold representation theorem, including its mathematical formulation and implications for overcoming dimensionality challenges. Next, we review the architecture and error-scaling properties of Kolmogorov-Arnold Networks, demonstrating how these networks achieve true freedom from the curse of dimensionality. Finally, we discuss the practical viability of Kolmogorov-Arnold Networks, highlighting scenarios where their unique capabilities position them to excel in real-world applications. This review aims to offer insights into Kolmogorov-Arnold Networks' potential to redefine scalability and performance in high-dimensional learning tasks.

Figures

Figures reproduced from arXiv: 2411.10622 by the authors.

Figure 1
Figure 1. Linear interpolation with 50 interpolant points - the linear interpolant struggles to capture the curvature of the function, resulting in a piecewise linear approximation. While linear interpolation is computationally efficient and provides a smooth curve with sufficient data points, it may not be ideal for accurately modeling functions with significant curvature. Interpolation is a technique used to estimate unknow… view at source ↗
Figure 2
Figure 2. Cubic spline interpolation with 50 interpolant points - The cubic spline interpolation notably improves the ability to capture the function’s curvature compared to linear interpolation, offering a smoother and more accurate fit [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. B-spline interpolation of order k = 3 with 50 interpolant points, using basis splines to approximate a cubic spline. The figure illustrates a basis spline with k = 5. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Basis functions used to perform the B-spline interpolation with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Different Types of interpolation performed with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Perceptron with bias x0 and inputs x1 and x2 along with their corresponding weights w0, w1 and w2. Although w0 has been shown explicitly here, often it is generalized as b. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: A simple [1,2,1] KAN layer illustrating the above mathematical representation. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Forward citations

Cited by 1 Pith paper

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

  1. FORTRESS: Function-composition Optimized Real-Time Resilient Structural Segmentation via Kolmogorov-Arnold Enhanced Spatial Attention Networks

    cs.CV 2025-07 reject novelty 4.0 of 10

    FORTRESS combines depthwise separable convolutions and a gated Kolmogorov-Arnold module to report F1 of 0.771 and mIoU of 0.677 on the CSDD benchmark, but the core KAN contribution is not isolated by ablation.

Reference graph

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