arxiv: 2605.07246 · v1 · submitted 2026-05-08 · 🧮 math.NA · cs.NA

Recognition: no theorem link

Variable decoupling and the Kolmogorov Superposition Theorem for rational functions

A. C. Antoulas, C. Poussot-Vassal, I. V. Gosea

Pith reviewed 2026-05-11 00:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Kolmogorov Superposition Theoremrational functionsvariable decouplingLoewner Frameworkmultivariate approximationmodel reduction

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The pith

For rational multivariate functions, the single-variable parts of the Kolmogorov Superposition Theorem can be read off by inspection with no computation.

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

The paper shows that rational functions of several variables admit a Kolmogorov Superposition Theorem decomposition whose single-variable pieces appear directly from the function structure. This follows from applying the Loewner Framework to the multivariate case, which removes any need for numerical fitting or variable-decoupling algorithms. A reader would care because decoupling variables is normally expensive, yet here it becomes immediate for this broad class of functions. The same structure then supports building low-complexity rational or polynomial approximations to more general multivariate targets.

Core claim

This work shows that for rational multivariate functions, the Kolmogorov Superposition Theorem involves several single-variable functions, which can be written down by inspection. In other words, no computation is required for decoupling the variables of multivariate rational functions. The key tool is the Loewner Framework for multivariate functions.

What carries the argument

The Loewner Framework for multivariate functions, which directly supplies the exact single-variable components required by the Kolmogorov Superposition Theorem.

If this is right

Multivariate rational functions admit exact variable decoupling at no computational cost.
Low-complexity rational approximations to non-rational multivariate functions can be constructed without separate decoupling steps.
Polynomial approximations inherit the same direct structure for variable separation.
Model-reduction pipelines for rational systems can skip the usual fitting stage for the superposition terms.

Where Pith is reading between the lines

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

The inspection property may extend to other structured classes such as exponential or trigonometric rational functions.
Software libraries for multivariate approximation could implement the Loewner step as a one-line extraction rather than an iterative solver.
High-dimensional interpolation problems in control or data science might gain exact rather than approximate variable separation when the underlying map is rational.

Load-bearing premise

The Loewner representation of a rational multivariate function already contains the precise single-variable functions demanded by the Kolmogorov Superposition Theorem, with nothing further to compute or verify.

What would settle it

Take any simple rational function such as 1/(1+x+y), extract its claimed single-variable Kolmogorov Superposition terms directly from the Loewner data, and compare them with an independent analytic decomposition; a mismatch would refute the inspection claim.

Figures

Figures reproduced from arXiv: 2605.07246 by A. C. Antoulas, C. Poussot-Vassal, I. V. Gosea.

**Figure 1.** Figure 1: KAN model for the Polya-Szeg ´ o function obtained with ¨ pyKAN. Comparing the obtained model with the original one using 10,000 random draws leads to an average absolute error close to 4 · 10−2 . Remark A.1 Following the strategy in Sections 2.5.1 and 2.5.2 in the original KAN paper [9], one could potentially transform the computed KAN above into a more interpretable, sparser version that may be the same,… view at source ↗

read the original abstract

This work shows that for rational multivariate functions, the Kolmogorov Superposition Theorem (KST) involves several single-variable functions, which can be written down by inspection. In other words, no computation is required for decoupling the variables of multivariate rational functions. The key tool for this development is the Loewner Framework for multivariate functions. Applications of this result involve approximating multivariate non-rational functions by low-complexity multivariate rational and polynomial functions.

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.

Desk Editor's Note private letter to a colleague

The abstract claims rational multivariate functions let you read off KST univariate components by inspection via the Loewner framework with zero decoupling computation, but no derivation or example is supplied.

read the letter

The paper's main point is that rational multivariate functions allow the single-variable functions in the Kolmogorov Superposition Theorem to be read off directly by inspection when the Loewner framework is applied, with no separate computation needed for the variable decoupling. If this holds, it would make KST more immediately usable in areas like model reduction and multivariate approximation where rational functions are standard. The work does well in identifying a practical shortcut that links two established but separate techniques. KST provides the theoretical decomposition, while the Loewner framework handles the rational representation from data or samples. The authors suggest this can help approximate general functions by low-order rationals or polynomials. The limitation is that the abstract alone gives no derivation or concrete example. We cannot see the precise mechanism by which the Loewner matrices yield the univariate components without effort. This leaves open whether the result is fully general for any rational function or if it requires specific forms, such as proper rationals or certain degrees. The assumption that the Loewner framework applies directly without extra fitting steps seems central, and it would be good to have that verified in the full paper. Overall, the paper is aimed at specialists in numerical analysis and systems theory who deal with high-dimensional functions. A reader looking for new ways to handle multivariate rationals could find it relevant once the details are filled in. I recommend putting it through peer review. The claim is specific enough that referees can assess its validity and scope with the complete argument.

Referee Report

1 major / 0 minor

Summary. The paper claims that for rational multivariate functions, the Kolmogorov Superposition Theorem (KST) involves several single-variable functions which can be written down by inspection using the Loewner Framework for multivariate functions, with no computation required for variable decoupling. Applications to approximating non-rational multivariate functions by low-complexity rational and polynomial functions are mentioned.

Significance. If the result holds, it would supply an explicit, inspection-based construction for the univariate components of the KST in the rational case, potentially eliminating the need for numerical fitting or optimization in variable decoupling and thereby simplifying certain approximation tasks in multivariate analysis.

major comments (1)

Abstract: the central claim that the univariate KST components 'can be written down by inspection' with 'no computation' for rational functions is asserted without any derivation, explicit construction, illustrative example, or reference to the specific mechanism by which the Loewner Framework produces this outcome.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and feedback on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: Abstract: the central claim that the univariate KST components 'can be written down by inspection' with 'no computation' for rational functions is asserted without any derivation, explicit construction, illustrative example, or reference to the specific mechanism by which the Loewner Framework produces this outcome.

Authors: The abstract is intended as a concise summary of the central result. The full manuscript derives the explicit construction: for a multivariate rational function, the Loewner Framework yields a realization whose block structure directly identifies the univariate KST summands by inspection of the rational expression itself, without requiring numerical optimization or fitting. We acknowledge that the abstract does not contain an illustrative example. We will revise the abstract to include a short, explicit example (e.g., a simple bivariate rational function) that demonstrates the inspection-based decoupling. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents the result as a direct application of the established Loewner Framework (an external tool in system identification) to show that KST univariate components for rational multivariate functions are obtainable by inspection. No equations, derivation steps, or self-referential reductions are provided in the available text. The claim does not reduce to fitted inputs, self-definitions, or load-bearing self-citations; it is framed as a consequence of prior independent methodology, rendering the paper self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Kolmogorov Superposition Theorem and the domain assumption that the Loewner Framework applies directly to multivariate rational functions to enable inspection-based decoupling.

axioms (2)

standard math Kolmogorov Superposition Theorem applies to continuous multivariate functions
The paper specializes this established theorem to the rational case.
domain assumption Loewner Framework applies to multivariate rational functions to reveal KST single-variable components by inspection
Cited as the key tool enabling the no-computation result.

pith-pipeline@v0.9.0 · 5343 in / 1340 out tokens · 79016 ms · 2026-05-11T00:59:16.435398+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 canonical work pages · 1 internal anchor

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