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arxiv: 2508.04392 · v2 · pith:QCDU63LDnew · submitted 2025-08-06 · 🧮 math.NA · cs.NA

Explicit Construction of Approximate Kolmogorov Superpositions with C2 Smoothness

Lunji Song , Zilan Cheng , Juan Diego Toscano , Li-Lian Wang This is my paper

Pith reviewed 2026-05-21 23:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Kolmogorov superpositionC2 smoothnessHolder continuous functionspiecewise interpolationshape functionsneural networksapproximation theoryexplicit construction
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The pith

Kolmogorov superpositions can be explicitly approximated by C2 smooth inner and outer functions to reach N to the power minus alpha accuracy for any alpha-Holder function.

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

The paper gives an explicit construction for approximate Kolmogorov superpositions that use only C2 smooth functions. The inner functions arise from translations and dilations of one fixed piecewise C2 strictly increasing function. The outer functions are assembled row by row via piecewise C2 interpolation that employs specially designed shape functions. With N outer terms the scheme approximates every alpha-Holder continuous function to within an error of order N to the minus alpha. Readers may care because the classical Kolmogorov functions are too irregular for direct use, while this version keeps the representation idea yet supplies the smoothness needed for many practical calculations.

Core claim

We explicitly construct an approximate version of the Kolmogorov superpositions, which is composed of C2-inner and outer functions, and can approximate an arbitrary alpha Holder continuous function with accuracy of N to the power -alpha, where N denotes the number of outer summations. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed rowwise through piecewise C2 interpolation using newly designed shape functions.

What carries the argument

Rowwise piecewise C2 interpolation of the outer functions via newly designed shape functions, together with translated and dilated copies of a single piecewise C2 strictly increasing function serving as the inner functions.

If this is right

  • The construction removes the pathological irregularity that normally appears in Kolmogorov superpositions.
  • The original Kolmogorov strategy of building multivariate functions from univariate ones is retained in approximate form.
  • The same explicit functions can be inserted directly into neural-network architectures that require C2 regularity.
  • The error bound scales as N to the minus alpha for any Holder exponent alpha between zero and one.

Where Pith is reading between the lines

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

  • Smooth Kolmogorov-type representations may now be substituted into existing numerical schemes that already demand twice-differentiable approximants.
  • The explicit shape-function construction suggests a template for obtaining higher-order smoothness versions of the same superposition by redesigning the interpolation pieces.
  • Because the paper already notes applicability to neural networks, the C2 property could be used to equip those networks with analytic derivatives for optimization or sensitivity analysis.

Load-bearing premise

The outer functions can be constructed rowwise through piecewise C2 interpolation using the newly designed shape functions while preserving the required approximation rate and C2 regularity.

What would settle it

Pick a concrete alpha-Holder function such as |x|^alpha on the unit interval, compute the constructed superposition for increasing N, and check whether the maximum error decreases exactly like N to the power minus alpha while the second derivatives of every outer function remain continuous across all knots.

Figures

Figures reproduced from arXiv: 2508.04392 by Juan Diego Toscano, Li-Lian Wang, Lunji Song, Zilan Cheng.

Figure 2.1
Figure 2.1. Figure 2.1: (a) Partition of [0, 1] by closed sub-intervals and gaps in (2.14) with N = J = 5 and δ = 1/25. (b)-(d) Plots of ψq(x) at Lq with q = 1, 2, 3. With the above setup, we define the corresponding Kolmogorov maps (see Brattka [3] for this concept). Definition 2.1 (Kolmogorov maps). Let λ = (λ1, . . . , λd) be a given sequence of positive numbers, and let ψq(t) be the piecewise C 2 - function defined in (2.16… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: (a)-(c) Illustration of disjoint hypercubes, “centers” and gaps at levels Lq with q = 1, 2, 3 in 2D. (d) Covering of [0, 1]2 at different levels by δ-shifting. Here, N = J = 5 and δ = 1/25. Correspondingly, we denote the mapped hypercube “centers” by a j q := Ψq(c j q ) = Ψq(c j q ;λ) = X d p=1 λpψq(c jp q ), 1 ≤ q ≤ N. (2.24) In the construction of outer functions, it is crucial to understand the distri… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Distributions of {a (k,j2) q } J−1 k=−1 for j2 = −1, . . . , J − 1 from bottom to top. Here, q = 1, N = J = 5, δ = 1/25 and µ = 1/2. Left: (λ1, λ2) ≈ (0.4772, 0.5228). Right: (λ1, λ2) ≈ (0.1321, 0.8679) which satisfies (2.31). We next show that the interlacing of a j q in different “rows” can be avoided for a special choice of λ (see [PITH_FULL_IMAGE:figures/full_fig_p010_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: (a) Surface plot of f(x1, x2) in (2.37). (b) Plots of the curves f(:, c j2 1 )/N with c j2 1 being the x2-coordinate of the “centers” as in [PITH_FULL_IMAGE:figures/full_fig_p013_2_4.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Convergence rate of the approximate superpositions. Errors against J = N (left), and J with N = 10 (right) in log-log scale. We next test higher dimensions and still take f(x) = u(x) ln(|u(x)|) with u(x) = tanh(ln(|sin(x1 + x2 + · · · + xd)| + ϵ) + e x1 tan(x2+···+xd) )/2 + X 5 i=1 αi e −(x1−β1,i) 2/σ2 i −(x2−β2,i) 2/σ2 i ···−(xd−βd,i) 2/σ2 i , (3.11) where we choose the constants so that |u(x)| can take… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Convergence rate of the approximate superpositions with J = N for d = 3 (left), d = 4 (middle) and d = 9 (right) in log-log scale. 3.3. Concluding remarks and discussions. It is well known that the univariate functions involved in the exact Kolmogorov–Arnold representation lack differentiability and smooth￾ness, often exhibiting “wild” behavior—even when the target function f(x) is smooth. This significa… view at source ↗
read the original abstract

We explicitly construct an approximate version of the Kolmogorov superpositions, which is composed of C2-inner and outer functions, and can approximate an arbitrary alpha Holder continuous function with accuracy of N to the power -alpha, where N denotes the number of outer summations. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed rowwise through piecewise C2 interpolation using newly designed shape functions. This novel variant of Kolmogorov superpositions overcomes the wild and pathological behaviors of the inherent single variable functions, but retains the essence of Kolmogorov strategy of exact representation-an objective that Sprecher (Neural Netw. 144(2021)438-442) has actively pursued. We also discuss the implications of this new construction and demonstrate its applicability to related neural networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper explicitly constructs an approximate version of Kolmogorov superpositions using C²-smooth inner and outer functions. Inner functions are obtained via translations and dilations of a fixed piecewise C² strictly increasing base function. Outer functions are built rowwise by piecewise C² interpolation with newly designed shape functions. The resulting superposition approximates arbitrary α-Hölder continuous functions with accuracy O(N^{-α}), where N is the number of outer summation terms. The construction is presented as overcoming pathological behaviors of classical Kolmogorov functions while retaining the superposition strategy, with discussion of implications for neural networks.

Significance. If the error analysis and regularity claims hold, the work supplies an explicit, C²-smooth realization of approximate Kolmogorov superpositions with a concrete rate for Hölder classes. This addresses a persistent difficulty in the field by replacing wild inner/outer functions with controllable smooth ones, potentially aiding both theoretical approximation results and practical neural-network constructions. The parameter-free character of the rate (no fitted constants or self-referential scaling) and the focus on explicit shape functions are strengths that would strengthen the contribution if fully verified.

major comments (1)
  1. [Outer function construction] Outer-function construction (as described following the inner-function definition): the claim that rowwise piecewise C² interpolation with the new shape functions simultaneously preserves global C² regularity and the overall N^{-α} rate for arbitrary α-Hölder targets is load-bearing. The local interpolation error must be shown to scale as O(h^α) (or better) with mesh size h chosen independently of N, without accumulation across the N terms or loss of C² matching at knots that would force h to depend on N. An explicit error bound relating the Hölder modulus, the shape-function properties, and the final approximation constant is required to substantiate the central theorem.
minor comments (3)
  1. [Abstract] The abstract states the approximation rate but does not specify the domain (e.g., [0,1]^d); adding this would improve precision.
  2. [Introduction] A short table comparing the smoothness, explicitness, and rate of the present construction with Sprecher (2021) and other recent variants would clarify the incremental advance.
  3. [Shape functions] Notation for the shape functions (e.g., their support, knot placement, and C² matching conditions) should be introduced with a small diagram or explicit formulas in the main text rather than deferred to an appendix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below, providing clarifications and indicating the revisions made.

read point-by-point responses

  1. Referee: Outer-function construction (as described following the inner-function definition): the claim that rowwise piecewise C² interpolation with the new shape functions simultaneously preserves global C² regularity and the overall N^{-α} rate for arbitrary α-Hölder targets is load-bearing. The local interpolation error must be shown to scale as O(h^α) (or better) with mesh size h chosen independently of N, without accumulation across the N terms or loss of C² matching at knots that would force h to depend on N. An explicit error bound relating the Hölder modulus, the shape-function properties, and the final approximation constant is required to substantiate the central theorem.

    Authors: We thank the referee for identifying this key point requiring greater explicitness. We agree that a fully detailed error analysis strengthens the presentation. In the revised manuscript we have added a dedicated subsection deriving the interpolation error bound. The analysis establishes that the local error of the rowwise piecewise C² interpolant scales as O(h^α) for any α-Hölder target, with mesh size h chosen independently of N. Global C² regularity is preserved because the newly designed shape functions enforce exact matching of function value, first derivative, and second derivative at every knot; these matching conditions depend only on the fixed properties of the shape functions and not on N. The overall superposition error is then shown to be O(N^{-α}) by combining the uniform bound on each outer function with the structure of the translated-dilated inner functions; no accumulation across the N terms occurs because each term’s contribution is controlled by the same N-independent constant. An explicit relation is now stated between the Hölder modulus of continuity, the supremum norms of the shape functions and their derivatives, and the constant appearing in the main theorem. These additions directly address the load-bearing claim without altering the construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction is self-contained

full rationale

The paper presents an explicit construction of approximate Kolmogorov superpositions using translated and dilated piecewise C2 inner functions together with rowwise piecewise C2 interpolation of outer functions via newly designed shape functions. The claimed N^{-alpha} rate for arbitrary alpha-Holder targets is asserted to follow directly from the approximation properties of these constructions. No equation reduces the rate or smoothness claim to a fitted parameter, prior self-citation, or definitional equivalence; the central steps rely on independent design choices whose error control is stated to be verified within the paper. The single external citation to Sprecher is not load-bearing for the derivation and does not import a uniqueness theorem or ansatz from the present authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard properties of Holder spaces and the ability to design C2 shape functions that support the interpolation while delivering the rate; no free parameters are introduced and no new physical entities are postulated.

axioms (1)
  • domain assumption Holder continuous functions admit approximation by sums of univariate compositions under suitable smoothness constraints on the components.
    Invoked to define the target class and the desired approximation rate.
invented entities (1)
  • Newly designed shape functions for piecewise C2 interpolation no independent evidence
    purpose: Enable rowwise construction of outer functions that remain C2 while supporting the N^{-alpha} error bound.
    Introduced in the outer-function construction without reference to prior literature.

pith-pipeline@v0.9.0 · 5675 in / 1386 out tokens · 35513 ms · 2026-05-21T23:30:56.304451+00:00 · methodology

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Reference graph

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