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REVIEW 4 major objections 6 minor 36 references

Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Fixed-phase sine sums approximate any continuous function.

desk verdict The SineKAN theorem is plausibly true, but the proof as written fails at Lemma 2, whose uniform error bound is false, and Lemma 4 has a phase-grid violation; the paper needs a substantive fix before it can be published. read the letter →

arxiv 2508.00247 v1 pith:6UWS4MB2 submitted 2025-08-01 stat.ML cs.LGcs.NAmath.NA

Sergei Gleyzer , Hanh Nguyen , Dinesh P. Ramakrishnan , Eric A. F. Reinhardt This is my paper
classification stat.MLcs.LGcs.NAmath.NA MSC 41A3068T0726B40
keywords Kolmogorov-ArnoldrepresentationuniversalapproximationsinusoidalactivationSineKANBernsteinpolynomialsFourierneuralnetworkarchitecture
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's goal is to show that the Kolmogorov-Arnold representation can be built out of sine functions alone: for any continuous function on $[0,1]$, a finite weighted sum of sines with learnable frequencies and fixed, linearly spaced phases can approximate the function uniformly to any prescribed accuracy. The same construction is then stacked into two layers to cover continuous multivariable functions on the unit cube. If the theorems hold, they supply a theoretical justification for a KAN variant whose activation is just sine, without splines or other basis functions. The numerical section reports that this sine-based network beats fixed-frequency Fourier truncation and is competitive with multilayer perceptrons on oscillatory test functions.

What carries the argument

The load-bearing object is the Taylor polynomial $T_N(\omega,\alpha,x)=\sum_{l=0}^N \omega^l \sin(\alpha+l\pi/2)x^l/l!$ of a sine term, together with the $(N+1)\times(N+1)$ matrix $M$ with entries $M_{lk}=\omega_k^l \sin(\alpha_k+l\pi/2)$. Lemma 4 asserts that, for suitable frequencies and for phases in $(0,\pi/2)$, $M$ can be made invertible, so solving $M A=(b_l l!)$ reproduces any polynomial $p(x)=\sum b_l x^l$ exactly as a sum of sine Taylor polynomials. Because Bernstein polynomials approximate any continuous function, and the sine Taylor remainder has an explicit bound, the proof chain reduces uniform approximation of $f$ to a linear algebra problem. The fixed phases $\alpha_k=k\alpha/(N+1)$ are the Lorentz-Sprecher device that leaves only frequencies and amplitudes free, and Theorem 2 composes such sine sums both inside and outside, using uniform continuity, to handle many variables.

What would settle it

A direct check: for $N=1,2,\dots,20$, compute the best uniform error for $f(x)=|x-1/2|$ using the fixed phases $\alpha_k=k\pi/(2(N+1))$; if the minimal error does not decrease to zero as $N$ grows, Theorem 1 is false. A weaker check that isolates the proof's machinery: evaluate the determinant of the $M$ matrix in Lemma 4 at these phases; a zero determinant for any $N$ shows the stated phase choice cannot reproduce arbitrary polynomials.

Watch

Extended reading notes

Core claim

The paper's central theorem states that with phases fixed to $\alpha_k = k\alpha/(N+1)$, for every continuous $f:[0,1]\to\mathbb{R}$ and every $\epsilon>0$ there is an $N_0$ such that for all $N>N_0$ some frequencies $\omega_k\in[0,2\pi]$ and amplitudes $A_k\in\mathbb{R}$ give $\sup_{[0,1]} |f(x)-\sum_{k=0}^N A_k\sin(\omega_k x+\alpha_k)|<\epsilon$. The theorem extends to continuous functions on the unit cube $[0,1]^n$: approximating each inner and outer function in the Kolmogorov-Arnold representation by such sine sums yields a two-layer sinusoidal network with the same universal approximation property. The paper also reports numerical experiments in which this sine-network construction outperforms fixed-frequency Fourier series and matches or beats multilayer perceptron baselines on several one- and two-dimensional functions with rapid oscillations and near singularities.

Load-bearing premise

The proof assumes that the error from replacing each sine by its Taylor polynomial can be made arbitrarily small even when the fitted amplitudes grow large, and that the fixed phase grid, including a zero phase, always allows the amplitudes $A_k$ to be solved for.

Editorial extensions

If this is right

  • If Theorem 1 is correct, a single hidden layer of sine units with learnable frequency and amplitude and a fixed phase grid is enough to approximate any continuous one-dimensional function on a compact interval.
  • If Theorem 2 is correct, continuous multivariable functions on the unit cube are approximable by two sinusoidal layers, so the same ansatz covers KAN-style architectures without splines or other basis functions.
  • The numerical comparisons imply that for oscillatory targets with near singularities, learning the frequencies rather than preselecting a Fourier grid gives better accuracy than truncated Fourier series at matched parameter counts.
  • The experiments also indicate that at matched parameter counts and FLOP estimates, the two-layer sine network is competitive with MLP baselines on the tested two-dimensional functions.

Reading between the lines

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

  • A consequence the paper leaves implicit: with phases fixed, each sine unit contributes only a frequency and an amplitude to the parameter count, so the model's capacity is easy to audit and trainability should hinge mainly on how the frequencies are initialized and updated.
  • The proof route via Bernstein polynomials suggests a natural testable refinement: for analytic targets the term count should scale with the decay of Taylor coefficients, while for functions with fractional regularity the convergence should slow, consistent with the paper's slow numerical convergence for $x^{1/5}\sin(1/x)$.
  • If the theorems hold, the representation transfers directly to periodic and time-series regression, where sinusoidal activations are a natural inductive bias; non-stationary sequence modeling is the obvious next testbed.
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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 / 6 minor

Summary. The manuscript proposes SineKAN, a Kolmogorov-Arnold network variant whose hidden activations are weighted sinusoids with learnable frequencies and fixed, linearly spaced phases. The main theoretical claims are Theorem 1, asserting that every continuous function on [0,1] can be approximated uniformly by a finite sum of such sinusoids, and Theorem 2, extending this to continuous multivariable functions via a two-layer sine network. The proof strategy is to approximate a continuous function by a Bernstein polynomial, represent that polynomial exactly as a linear combination of Taylor polynomials of shifted sinusoids, and then use Taylor remainder bounds to replace the Taylor polynomials by the sinusoids themselves. The manuscript also reports numerical comparisons against Fourier series and MLPs on several test functions and provides a link to a code repository.

Significance. If the theorems were proved, they would provide a clean and constructive universal approximation result for a constrained one-hidden-layer sinusoidal architecture, with a fixed linear phase schedule reducing the number of free parameters. The public code repository and the numerical comparison against Fourier and MLP baselines are positive features. However, the proof of the central theorem depends on a lemma that is false as stated and on an unproved matrix-invertibility assertion, and the argument supplies no control on the amplitudes produced by the linear system. Because Theorem 2 inherits the defects of Theorem 1, both theoretical claims are currently unsupported. The significance is therefore conditional on a substantial revision of the proofs.

major comments (4)
  1. [Section 2, Lemma 2] Lemma 2 is false as stated. The final displayed bound is (2π)^(N+1)/N! times max_k |A_k|, and since max_k |A_k| is unrestricted, this cannot be made smaller than ε for all choices. Explicitly, for any proposed N0 take N = N0+1, A0 = M, A_k = 0 for k ≥ 1, ω0 = 2π, α0 = π/4. Then the left side of (4) equals M · sup_x |sin(2πx + π/4) − T_N(2π, π/4, x)|, which is positive because a sine is not a degree-N polynomial; choosing M larger than ε divided by that sup violates (4). Thus the 'for all A_k ∈ R' quantifier in the lemma cannot hold.
  2. [Section 2, Lemma 4] Lemma 4 asserts that for any polynomial p and any α_k ∈ (0, π/2) one can choose ω_k ∈ [0,2π] making the matrix M invertible, but the proof only says 'by induction in N, we can select...' and provides no inductive construction, determinant formula, or lower bound. This is a load-bearing unproved assertion. Moreover, Theorem 1 sets α_k = kα/(N+1), so α_0 = 0; the hypothesis α_k ∈ (0, π/2) of Lemma 4 fails for k = 0, and the claim that 'sin(α_k + lπ/2) ≠ 0' is void for α_0 = 0 when l is even. Consequently Lemma 4 cannot be invoked as stated in the proof of Theorem 1.
  3. [Section 2, proof of Theorem 1] The ε/2 combination in equations (8) and (9) is not justified. Lemma 3 gives a Bernstein polynomial p with |f − p| < ε/2, and Lemma 4 gives p = Σ A_k T_N(ω_k, α_k, x). Lemma 2, even if repaired, would require the amplitudes A_k produced by Lemma 4 to satisfy a uniform bound independent of N, or at least a bound compatible with the chosen N. Lemma 4 provides no bound on the solution A of (7), and the entries of that system depend on the chosen ω_k and α_k. Without an amplitude bound, or an alternative argument showing that the Taylor error tends to zero for the specific A_k selected, the triangle inequality leading to (2) does not follow.
  4. [Section 3, proof of Theorem 2] The proof applies Theorem 1 to the functions ψ_pq on [0,1] and then to φ_q on [a,b], but Theorem 1 is stated only for continuous functions on [0,1]. The proof omits the required affine rescaling for the outer functions and for the composite variable y = Σ S_pq, which ranges over [a,b], not [0,1]. Additionally, the theorem statement leaves N and M as free parameters rather than quantifying their existence, and the proof does not specify how the index ranges for q and j interact with the error decomposition in (12)–(13). The claimed total error bound is therefore incomplete.
minor comments (6)
  1. [Section 2, proof of Lemma 2] The final sentence says 'to complete the proof of Lemma 3,' but the lemma being proved is Lemma 2; this cross-reference should be corrected.
  2. [Theorem 2 statement] The statement contains the typo 'The for any ε > 0'; it should read 'Then for any ε > 0'.
  3. [Section 4, equations (15)–(17)] The architecture description uses phases k/(G+1) and k/(G+1) + lπ/(N+1), whereas the proved Theorem 1 uses kα/(N+1) and Theorem 2 uses a fixed phase schedule; the relation between the implemented phase schedule and the theorem should be clarified or aligned.
  4. [Figures 1 and 2] The manuscript text provides only figure captions; the actual plots are not present in the submitted text, which prevents verification of the reported numerical comparisons. Please include the figures.
  5. [References] References [26] and [27] are duplicate entries for the same Rumelhart, Hinton, and Williams (1986) paper and should be merged or disambiguated.
  6. [Section 4, experimental setup] The statement that each function is fitted for a default of 100 steps per fitted parameter is ambiguous, as it does not specify the number of optimization iterations, the parameter initialization range, or the selection criterion for the grid size in Figure 1; these details should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are attempted proofs from Weierstrass and Taylor expansion, not reductions to their own inputs; the identified defects are technical gaps, not circular reasoning.

full rationale

The paper's derivation chain is a genuine constructive attempt: Theorem 1 is proved via Lemmas 1-4, which approximate sine by Taylor polynomials, represent Bernstein polynomials as sums of those Taylor polynomials, and then combine the two approximations. This does not assume the conclusion of Theorem 1, nor does it fit parameters and then call the fit a prediction. The main defects identified by the skeptic are mathematical errors, not circularity: Lemma 2's uniform error bound is false because max_k |A_k| can be arbitrarily large and may depend on N, and Lemma 4's invertibility condition requires all alpha_k in (0, pi/2) while Theorem 1 sets alpha_0 = 0. These are correctness gaps in the proof, but they do not make the argument circular. The numerical experiments fit the proposed SineKAN model to test functions and compare errors against baselines; they do not present fitted outputs as theoretical predictions. The only self-citation, reference [24], is used for motivation and comparison in the introduction and discussion, and it is not load-bearing for the proof of Theorem 1 or Theorem 2. No uniqueness theorem from the authors is invoked, no ansatz is smuggled in via citation, and no known result is merely renamed. The proofs rest on external results (Weierstrass approximation, Kolmogorov-Arnold representation) and the paper's own lemmas, so the derivation is self-contained in the sense relevant to circularity analysis.

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

No new physical entities or parameters fitted to data are introduced. The paper's central claim rests on two unproven technical assumptions (matrix invertibility and coefficient growth control), which are listed as ad hoc axioms.

assumptions (4)
  • ad hoc to paper The determinant of the matrix M in Lemma 4 is nonzero for some choice of omega_0,...,omega_N in [0,2pi]
    The proof asserts this by induction without providing the induction argument; the claim is essential for representing an arbitrary polynomial as a sine Taylor sum. It is not established for the specific phases alpha_k=k alpha/(N+1), where alpha_0=0 violates the lemma's hypothesis.
  • ad hoc to paper The amplitudes A_k in Lemma 2 can be treated as uniformly bounded in N, or at least grow slowly enough that the Taylor remainder bound tends to zero
    Lemma 2 is only true if max|A_k| is controlled, but the proof quantifies over all A_k, and the A_k from Lemma 4 are not bounded. The proof of Theorem 1 silently assumes this control.
  • standard math Weierstrass approximation theorem
    Invoked in Lemma 3 to approximate continuous f by Bernstein polynomials.
  • standard math Kolmogorov-Arnold representation theorem
    Invoked in Theorem 2 to reduce multivariable approximation to 1D inner and outer functions; if the theorem's scope is restricted to the unit cube it applies.

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Cite this review

Pith. "Pith review of Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks." pith.science (2026). https://pith.science/paper/6UWS4MB2

@misc{pith2026250800247,
  author       = {Pith},
  title        = {Pith review of: Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UWS4MB2}},
  note         = {Machine review of arXiv:2508.00247}
}
read the original abstract

The Kolmogorov-Arnold representation theorem states that any continuous multivariable function can be exactly represented as a finite superposition of continuous single variable functions. Subsequent simplifications of this representation involve expressing these functions as parameterized sums of a smaller number of unique monotonic functions. These developments led to the proof of the universal approximation capabilities of multilayer perceptron networks with sigmoidal activations, forming the alternative theoretical direction of most modern neural networks. Kolmogorov-Arnold Networks (KANs) have been recently proposed as an alternative to multilayer perceptrons. KANs feature learnable nonlinear activations applied directly to input values, modeled as weighted sums of basis spline functions. This approach replaces the linear transformations and sigmoidal post-activations used in traditional perceptrons. Subsequent works have explored alternatives to spline-based activations. In this work, we propose a novel KAN variant by replacing both the inner and outer functions in the Kolmogorov-Arnold representation with weighted sinusoidal functions of learnable frequencies. Inspired by simplifications introduced by Lorentz and Sprecher, we fix the phases of the sinusoidal activations to linearly spaced constant values and provide a proof of its theoretical validity. We also conduct numerical experiments to evaluate its performance on a range of multivariable functions, comparing it with fixed-frequency Fourier transform methods and multilayer perceptrons (MLPs). We show that it outperforms the fixed-frequency Fourier transform and achieves comparable performance to MLPs.

Figures

Figures reproduced from arXiv: 2508.00247 by the authors.

Figure 1
Figure 1. Approximation error as a function of grid size: top left eq. (18), middle left eq. (19), [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Loss as a function of number of parameters (Left) and FLOPs (Right) for eq. (23) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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