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REVIEW 2 major objections 2 minor 72 references

Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks

T0 review · 2 major / 2 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read The Radon-domain L^p space of ridge integrals recovers the critical Sobolev space H^{k+(d+1)/2} exactly when p=2.

desk verdict The paper defines a Radon-domain L^p space for ReLU^k ridge integrals, proves it recovers the critical Sobolev space at p=2 via Fourier analysis, and extracts explicit approximation rates from that link. read the letter →

arxiv 2606.24795 v1 pith:SKK7L7BJ submitted 2026-06-23 math.NA cs.NA

Juncai He , Zitong Tian This is my paper · ORCID
classification math.NAcs.NA
keywords RadontransformSobolevspacesridgefunctionsReLUnetworksapproximationratesFourieranalysisneuralnetwork
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 defines the Radon-domain L^p space R L^p_k(Ω) as the set of functions on a bounded domain that admit a ridge integral representation whose coefficient density lies in L^p of the Radon domain. It proves that this space coincides with the Sobolev space H^{k+(d+1)/2}(Ω) for p=2 by direct Fourier analysis. For 1

What carries the argument

The Radon-domain L^p space R L^p_k(Ω) consisting of ridge-integral representations whose coefficient densities belong to L^p in the Radon domain; the space encodes the combined effect of activation regularity k and Radon back-projection on Sobolev regularity.

What would settle it

Exhibit a concrete function in H^{k+(d+1)/2}(Ω) whose Radon-domain coefficient density fails to belong to L^2, or compute the exact Sobolev index shift produced by the Radon transform multiplier and check whether it matches the Seeger-Sogge-Stein amount.

Watch

Extended reading notes

Core claim

The Radon-domain L^p space R L^p_k(Ω) recovers the critical Sobolev space H^{k+(d+1)/2}(Ω) for p=2 by elementary Fourier analysis, while for 1<p<∞ it forms a Sobolev sandwich whose gap on each side equals the Seeger-Sogge-Stein loss for the Radon transform.

Load-bearing premise

That every function in the target Sobolev space admits a ridge integral representation whose coefficient density lies in the required L^p space on the Radon domain.

Editorial extensions

If this is right

  • The identification yields the optimal Hilbert-space approximation rate O(n^{-1/2 - (2k+1)/(2d)}) for linearized ReLU^k networks at p=2.
  • Discretization of the ridge integral by a deterministic interpolation skeleton plus uniform sampling produces high-probability L^p approximation rates for any 1<p<∞.
  • The joint action of the activation power k and the Radon-transform loss fixes the precise Sobolev regularity attainable by the network class.

Reading between the lines

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

  • The same Radon-domain construction could be applied to other integral representations to obtain Sobolev characterizations for networks with different activations.
  • Iterating the ridge-integral representation might give analogous sandwich results for deeper networks.
  • The explicit loss term suggests that quadrature rules adapted to the Radon geometry could improve practical training rates beyond generic sampling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Radon-domain L^p space ΡL^p_k(Ω) consisting of functions on a bounded domain Ω that admit a ridge-integral representation with L^p coefficient density in the Radon domain. It claims that for p=2 this space coincides exactly with the critical Sobolev space H^{k+(d+1)/2}(Ω) via elementary Fourier analysis, while for 1<p<∞ the space is sandwiched between two Sobolev spaces whose gap on each side equals the Seeger–Sogge–Stein loss of the Radon transform as a Fourier integral operator. The paper further discretizes the representation via deterministic interpolation and uniform sampling to obtain high-probability L^p approximation rates, including the optimal rate O(n^{-1/2-(2k+1)/(2d)}) for linearized networks at p=2.

Significance. If the identifications and rates are rigorously established, the work supplies a sharp functional-analytic characterization of the approximation spaces realized by shallow ReLU^k networks, directly tying activation smoothness to Sobolev regularity through the Radon back-projection. The explicit discretization yielding optimal Hilbert-space rates constitutes a concrete, falsifiable contribution.

major comments (2)
  1. [§3] §3 (Hilbert-space identification): The claimed equality ΡL^2_k(Ω) = H^{k+(d+1)/2}(Ω) is asserted via global Fourier analysis and the Fourier-slice theorem for the Radon transform, yet the manuscript supplies no explicit extension operator E: H^{k+(d+1)/2}(Ω) o H^{k+(d+1)/2}(R^d) that preserves the existence of an L^2 Radon-domain density. Without such an operator the restriction argument does not close, and the equivalence may fail or acquire boundary-dependent constants.
  2. [§4] §4 (Sobolev sandwich): The statement that the gap on each side of the sandwich is exactly the Seeger–Sogge–Stein loss is load-bearing for the sharpness claim, but the proof sketch does not verify that the Radon-domain L^p norm controls the precise loss term after localization to the bounded domain Ω; a concrete estimate relating the two norms is required.
minor comments (2)
  1. [Introduction] Notation: the symbol ΡL^p_k(Ω) is introduced without an explicit comparison table to the classical ridge-function spaces used in the neural-network literature; adding one would improve readability.
  2. [Discretization section] The discretization argument in the final section invokes uniform sampling but does not state the precise probability space or the measure on the Radon domain; a short paragraph clarifying the sampling measure would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. Both points identify places where the manuscript would benefit from additional explicit constructions and estimates; we will incorporate these in the revision.

read point-by-point responses

  1. Referee: [§3] §3 (Hilbert-space identification): The claimed equality ΡL^2_k(Ω) = H^{k+(d+1)/2}(Ω) is asserted via global Fourier analysis and the Fourier-slice theorem for the Radon transform, yet the manuscript supplies no explicit extension operator E: H^{k+(d+1)/2}(Ω) → H^{k+(d+1)/2}(R^d) that preserves the existence of an L^2 Radon-domain density. Without such an operator the restriction argument does not close, and the equivalence may fail or acquire boundary-dependent constants.

    Authors: We agree that an explicit extension operator must be supplied to close the argument rigorously. In the revised manuscript we will insert a dedicated paragraph (or short subsection) constructing a bounded linear extension E: H^{k+(d+1)/2}(Ω) → H^{k+(d+1)/2}(R^d) that is compatible with the Fourier-slice theorem; one standard choice is the Stein extension operator (or a reflection-based extension when Ω is Lipschitz), whose Fourier multiplier properties ensure that the Radon-domain L^2 density remains in L^2 after extension. With this operator the global Fourier identification on R^d restricts correctly to Ω, yielding the claimed equality with constants independent of the particular extension. revision: yes

  2. Referee: [§4] §4 (Sobolev sandwich): The statement that the gap on each side of the sandwich is exactly the Seeger–Sogge–Stein loss is load-bearing for the sharpness claim, but the proof sketch does not verify that the Radon-domain L^p norm controls the precise loss term after localization to the bounded domain Ω; a concrete estimate relating the two norms is required.

    Authors: We accept that a concrete localization estimate is required. In the revision we will add an explicit lemma that, for a smooth cutoff χ supported in a neighborhood of Ω, relates the Radon-domain L^p norm of χf to the Seeger–Sogge–Stein loss term plus a controllable remainder. The proof proceeds by writing the localized Radon transform as a Fourier integral operator, applying the known global loss bounds, and estimating the commutator terms arising from the cutoff via integration by parts and the smoothness of the Radon kernel; the resulting constants depend only on the diameter of Ω and the support of χ, thereby confirming that the sandwich gap is precisely the Seeger–Sogge–Stein loss after localization. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Fourier identification and rates are self-contained

full rationale

The paper's central identification of RL^2_k(Ω) with H^{k+(d+1)/2}(Ω) is obtained directly via elementary Fourier analysis applied to the ridge-integral representation; the Sobolev sandwich for 1<p<∞ invokes the external Seeger-Sogge-Stein loss for the Radon transform as a Fourier integral operator. Neither step reduces to a fitted parameter, self-definition, or self-citation chain. The approximation rates follow from explicit discretization of the integral representation via deterministic interpolation and uniform sampling, producing concrete high-probability bounds without circular reduction to the input data or definitions. The analysis is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The paper introduces one new space and relies on two standard mathematical facts; no numerical parameters are fitted and no new physical entities are postulated.

assumptions (2)
  • standard math Elementary Fourier analysis suffices to recover the Sobolev regularity from the ridge integral representation
    Invoked to prove the p=2 case recovers H^{k+(d+1)/2}(Ω)
  • domain assumption The Radon transform acts as a Fourier integral operator whose loss is given exactly by the Seeger-Sogge-Stein theorem
    Used to quantify the sharp gap on each side of the Sobolev sandwich for 1<p<∞
invented entities (1)
  • Radon-domain L^p space R L^p_k(Ω)
    purpose: Space of functions admitting ridge integral representations with L^p coefficient density in the Radon domain
    Newly defined central object whose properties are proved in the paper

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Cite this review

Pith. "Pith review of Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks." pith.science (2026). https://pith.science/paper/SKK7L7BJ

@misc{pith2026260624795,
  author       = {Pith},
  title        = {Pith review of: Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SKK7L7BJ}},
  note         = {Machine review of arXiv:2606.24795}
}
abstract

We develop the $L^p$ space and approximation theory for shallow neural networks with $\mathrm{ReLU}^k$ activations. The central object is the Radon-domain $L^p$ space $\mathcal{R}L^p_k(\Omega)$ containing all functions on a bounded domain $\Omega$ that admit a ridge integral representation whose coefficient density belongs to $L^p$ in the Radon domain. In the Hilbert case $p=2$, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space $H^{k+(d+1)/2}(\Omega)$. For general $1<p<\infty$, the identity becomes a Sobolev sandwich. The sharp gap of each side is exactly the Seeger--Sogge--Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability $L^p$ approximation rates and the optimal Hilbert rate $O\!\big(n^{-\frac12-\frac{2k+1}{2d}}\big)$ at $p=2$ for linearized neural networks.

Figures

Figures reproduced from arXiv: 2606.24795 by the authors.

Figure 5.1
Figure 5.1. Convergence curves for k = 3 and mixed ratio 0.5. For each requested neuron size, the curve is plotted at the median post-filter number of active neurons, and the vertical value is the median relative error over thirteen seeds. Dashed black segments show the theoretical slope M−sk/d , sk = (d + 2k + 1)/2. mixed sampler stays between these two baselines in four of the five dimensions while still keeping the random co… view at source ↗

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Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.