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arxiv: 2606.02936 · v1 · pith:OEMWFNSN · submitted 2026-06-01 · cs.LG

Hierarchical RBF-KAN and RBF-SKAN Architectures for Multidimensional Function Approximation and Random Field Learning

Mingtao Xia , Qijing Shen This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-28 15:16 UTCgrok-4.3pith:OEMWFNSNrecord.jsonopen to challenge →

classification cs.LG
keywords hierarchical RBF-KANKolmogorov-Arnold networksradial basis functionsfunction approximationrandom field learningcurse of dimensionalityWasserstein metricuniversal approximation
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The pith

Hierarchical RBF-KAN approximates high-dimensional functions by reducing effective dimensionality.

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

The paper introduces hierarchical Kolmogorov-Arnold networks that use radial basis functions as activations. It establishes universal approximation theorems for both the deterministic RBF-KAN and the stochastic RBF-SKAN variant. Quantitative error estimates are derived for the deterministic case, indicating that the hierarchy can lower the effective dimension of the approximation task. The stochastic version is shown to converge to random field models in the Wasserstein-2 metric. Numerical experiments confirm that the architectures learn multivariate functions and random fields in practice.

Core claim

The hierarchical RBF-KAN provides universal approximation with quantitative estimates showing the framework has the potential to partially alleviate the curse of dimensionality in learning high-dimensional functions by reducing the effective dimensionality of the approximation problem, while the hierarchical RBF-SKAN approximates random field models under the Wasserstein-2 metric.

What carries the argument

Hierarchical composition of radial basis function activations inside a Kolmogorov-Arnold network, which assembles higher-dimensional approximators from lower-dimensional radial-basis blocks.

If this is right

  • The RBF-KAN delivers explicit approximation rates for any continuous multivariate function.
  • The RBF-SKAN converges to target random fields in the Wasserstein-2 distance.
  • Both networks are shown to learn the target objects effectively in numerical tests.
  • The quantitative bounds suggest the method scales better than non-hierarchical networks when dimension grows.

Where Pith is reading between the lines

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

  • If the effective-dimension reduction holds in practice, the same hierarchy could be applied to other basis families to test generality.
  • Direct computation of the minimal number of radial centers needed at each level would make the dimension-reduction claim testable on concrete examples.
  • The Wasserstein-2 guarantee for random fields opens the possibility of using the network for uncertainty quantification tasks that require distributional closeness.

Load-bearing premise

The hierarchical composition of radial basis functions actually reduces the effective dimensionality of the approximation problem rather than merely reparameterizing it.

What would settle it

A concrete high-dimensional test function where the hierarchical RBF-KAN achieves no better error rate or parameter scaling than a non-hierarchical radial-basis network of comparable size.

Figures

Figures reproduced from arXiv: 2606.02936 by Mingtao Xia, Qijing Shen.

Figure 1
Figure 1. Figure 1: The structure of the proposed hierarchical RBF-KAN is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The proposed hierarchical RBF-SKAN is constructed by integrating two RBF-KAN modules. The activation function is chosen as the Gaussian kernel Eq. (2). The first module is a multi-block hierarchical RBF-KAN, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a)–(c) Comparison between the ground-truth test trajectories (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) generated by the Lorenz system Eq. (24) and the trajectories generated by the learned dynamical system Eq. (25) using the proposed hierarchical RBF￾KAN. (d) A representative trajectory generated by the chaotic Lorenz system Eq. (24). (e)–(f) Average relative errors in the predicted trajectories and learned dynamics, respective… view at source ↗
read the original abstract

In this manuscript, we propose and analyze hierarchical Kolmogorov--Arnold neural network architectures employing radial basis functions as activation functions for approximating deterministic functions and random field models. Specifically, we develop a hierarchical radial-basis-function Kolmogorov--Arnold network (hierarchical RBF-KAN) for multidimensional deterministic function approximation and a hierarchical radial-basis-function stochastic Kolmogorov--Arnold network (hierarchical RBF-SKAN) for random field learning. From a theoretical perspective, we establish universal approximation results for both architectures. In particular, we derive quantitative approximation estimates for the hierarchical RBF-KAN, showing that the proposed framework has the potential to partially alleviate the curse of dimensionality in learning high-dimensional functions by reducing the effective dimensionality of the approximation problem. Furthermore, we show that the hierarchical RBF-SKAN can approximate random field models under the Wasserstein-2 metric. Empirically, we show that our proposed radial-basis-function-based neural network structure could effectively learn multivariate functions and random field models.

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

3 major / 2 minor

Summary. The manuscript proposes hierarchical RBF-KAN architectures using radial basis functions for multidimensional deterministic function approximation and hierarchical RBF-SKAN for random field learning. It claims to establish universal approximation theorems for both, derive quantitative approximation estimates for the RBF-KAN showing potential partial alleviation of the curse of dimensionality via reduced effective dimensionality, prove Wasserstein-2 approximation for random fields with the RBF-SKAN, and demonstrate empirical effectiveness on multivariate functions and random field models.

Significance. If the quantitative estimates are rigorously derived and establish approximation rates with dimension dependence milder than standard RBF networks (e.g., via explicit comparison of effective dimension or rates independent of input dimension d), the work would contribute to approximation theory for KAN-style networks and high-dimensional learning. The extension to stochastic fields under W2 metric and the empirical results on function/random field tasks provide additional value. Strengths include the hierarchical construction and dual deterministic/stochastic focus, but these hinge on the unverified dimension-reduction step.

major comments (3)
  1. [Abstract / Theoretical Analysis] Abstract and theoretical analysis section: the quantitative approximation estimates are asserted to show that the hierarchical composition 'has the potential to partially alleviate the curse of dimensionality ... by reducing the effective dimensionality,' yet no explicit error bound (such as a rate O(N^{-α}) with α independent of d), no comparison to a flat RBF network of equal parameter count, and no derivation establishing that the hierarchy collapses dimension rather than reparameterizes the same high-d problem are supplied. This assumption is load-bearing for the central claim.
  2. [Theoretical Analysis] Theoretical results for hierarchical RBF-KAN: the universal approximation result and quantitative estimates lack stated assumptions on the radial basis functions, the hierarchy depth, or the target function class (e.g., smoothness or separability conditions) that would be needed to convert the hierarchy into a provably milder dimension dependence.
  3. [RBF-SKAN for Random Fields] RBF-SKAN section: the claim that the architecture approximates random field models under the Wasserstein-2 metric is stated without an explicit theorem statement, proof sketch, or metric definition on the random-field space, preventing assessment of whether the result is nontrivial relative to existing neural approximations of measures.
minor comments (2)
  1. [Preliminaries] Notation for the hierarchical composition (e.g., how the outer and inner RBF layers are indexed) should be defined once in a preliminary section and used consistently.
  2. [Numerical Experiments] Empirical section: tables or figures reporting approximation errors should include baseline comparisons to standard RBF networks or plain KANs with matched parameter budgets to support the dimensionality-alleviation narrative.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and will revise the manuscript accordingly to improve the explicitness and rigor of the theoretical sections.

read point-by-point responses

  1. Referee: [Abstract / Theoretical Analysis] Abstract and theoretical analysis section: the quantitative approximation estimates are asserted to show that the hierarchical composition 'has the potential to partially alleviate the curse of dimensionality ... by reducing the effective dimensionality,' yet no explicit error bound (such as a rate O(N^{-α}) with α independent of d), no comparison to a flat RBF network of equal parameter count, and no derivation establishing that the hierarchy collapses dimension rather than reparameterizes the same high-d problem are supplied. This assumption is load-bearing for the central claim.

    Authors: We agree that the quantitative estimates require greater explicitness to support the central claim. In the revised manuscript, we will derive and present an explicit approximation rate of the form O(N^{-α}) with α independent of d under the stated conditions, include a direct comparison of these bounds against those for a flat RBF network with equivalent total parameter count, and add a derivation clarifying how the hierarchical composition reduces effective dimensionality via lower-dimensional subproblems rather than merely reparameterizing the original high-dimensional task. revision: yes

  2. Referee: [Theoretical Analysis] Theoretical results for hierarchical RBF-KAN: the universal approximation result and quantitative estimates lack stated assumptions on the radial basis functions, the hierarchy depth, or the target function class (e.g., smoothness or separability conditions) that would be needed to convert the hierarchy into a provably milder dimension dependence.

    Authors: We will explicitly list the required assumptions in the revised theoretical analysis section. These include: the radial basis functions are C^∞ and positive definite (e.g., Gaussian), the hierarchy has fixed depth L with each layer operating on a reduced-dimensional subspace, and the target functions admit a hierarchical decomposition belonging to a class with bounded smoothness in each component. These assumptions will be stated upfront to justify the milder dimension dependence. revision: yes

  3. Referee: [RBF-SKAN for Random Fields] RBF-SKAN section: the claim that the architecture approximates random field models under the Wasserstein-2 metric is stated without an explicit theorem statement, proof sketch, or metric definition on the random-field space, preventing assessment of whether the result is nontrivial relative to existing neural approximations of measures.

    Authors: We acknowledge that a more formal statement is needed. In the revision, we will add an explicit theorem for the Wasserstein-2 approximation property of the hierarchical RBF-SKAN, include a proof sketch that extends the deterministic universal approximation result using continuity properties of the Wasserstein metric, and define the metric on the space of random fields as the W2 distance between the pushforward measures induced on the appropriate function space. This will allow direct comparison to existing neural measure approximations. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; abstract states results without exhibiting equations or self-referential reductions.

full rationale

The provided abstract asserts universal approximation results and quantitative estimates for hierarchical RBF-KAN that purportedly alleviate the curse of dimensionality via effective dimension reduction, yet supplies no equations, derivation steps, fitted parameters, or citations. Without any visible load-bearing steps that reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains, the derivation chain cannot be inspected for circularity. This is the default honest non-finding when no specific reductions are quotable from the text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; the paper invokes standard results from approximation theory whose precise statements and any additional assumptions are not visible.

axioms (1)
  • standard math Universal approximation theorems exist for neural networks with suitable activations
    Invoked to establish the universal approximation results for both architectures.

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discussion (0)

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