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CS-422 · computer science · mathematics · Computer science

Pre-Asymptotic Sampling Numbers of Barron Spaces

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A sampling number is the smallest worst-case reconstruction error achievable from $m$ point evaluations, with the admissible linear or nonlinear reconstruction convention fixed. A Barron space controls a neural-integral representation norm; Fourier-analytic spaces instead constrain weighted Fourier content.

Statement recorded in the supplied source.
Determine the asymptotic nonlinear sampling numbers for variational Barron spaces. For the Fourier-analytic spaces $U_d^\alpha$, determine the pre-asymptotic dependence of $\sigma_m(U_d^\alpha)_{L^p}$ on $d$ and $m$, and determine the optimal reconstruction error when the sample points are uniform random points. [S2]
Scope qualification.
The norms defining the variational Barron space and $U_d^\alpha$, the sampling domain, and the error/probability convention are source-specific; pre-asymptotic dimension dependence is part of the target.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Determine optimal errors from finitely many samples in the stated Barron and Fourier-analytic function spaces, including random samples.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005896 (OWR-14298366-008). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Applied Harmonic Analysis and Data Science (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/21, pp. 1163--1226 https://doi.org/10.4171/owr/2024/21.
  • S3 Voigtlaender et al., Barron sampling numbers https://arxiv.org/abs/2208.07605. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30005896 and source code OWR-14298366-008 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-422-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-422

Pre-Asymptotic Sampling Numbers of Barron Spaces

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A sampling number is the smallest worst-case reconstruction error achievable from $m$ point evaluations, with the admissible linear or nonlinear reconstruction convention fixed. A Barron space controls a neural-integral representation norm; Fourier-analytic spaces instead constrain weighted Fourier content.

Statement recorded in the supplied source.
Determine the asymptotic nonlinear sampling numbers for variational Barron spaces. For the Fourier-analytic spaces $U_d^\alpha$, determine the pre-asymptotic dependence of $\sigma_m(U_d^\alpha)_{L^p}$ on $d$ and $m$, and determine the optimal reconstruction error when the sample points are uniform random points. [S2]
Scope qualification.
The norms defining the variational Barron space and $U_d^\alpha$, the sampling domain, and the error/probability convention are source-specific; pre-asymptotic dimension dependence is part of the target.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Determine optimal errors from finitely many samples in the stated Barron and Fourier-analytic function spaces, including random samples.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005896 (OWR-14298366-008). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Applied Harmonic Analysis and Data Science $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/21, pp. 1163--1226 https://doi.org/10.4171/owr/2024/21.
  • S3 Voigtlaender et al., Barron sampling numbers https://arxiv.org/abs/2208.07605. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30005896 and source code OWR-14298366-008 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A dimension-explicit nullspace obstruction for the actual Fourier class

The primary report defines its Barron unit ball using an integrable Fourier density, not an arbitrary discrete Fourier measure. Bare sine waves therefore cannot simply be inserted as admissible functions. We repair that attempted construction by a Gaussian envelope. Work on $\Omega_d=[-1/2,1/2]^d$, $d\ge1$, with smoothness $\alpha=1$ and Euclidean frequency norm. The unit ball consists of real restrictions admitting a density $F$ with $\int(1+|\xi|)|F(\xi)|\,d\xi\le1$.
Proposition: Every deterministic algorithm using $m<d$ point values, even adaptively, has worst-case $L^2(\Omega_d)$ error at least \[ \frac{e^{-\pi/4}}{3\sqrt{2d}} \] on this unit ball.
Proof: Put $\phi_d(x)=e^{-\pi|x|^2/d}$. Its Fourier density under the $2\pi$ convention is $G_d(\xi)=d^{d/2}e^{-\pi d|\xi|^2}$, with integral one and second moment $\int|\xi|^2G_d=1/(2\pi)$. The one-dimensional transform identity follows by differentiating the Fourier integral, integrating by parts to obtain its first-order Gaussian differential equation, and using the Gaussian integral at zero; tensor products give the stated $d$-dimensional formula and moments. For $a\in\mathbb R^d$ with $\|a\|_1=1$, define \[ f_a(x)=\frac{\phi_d(x)}3\sum_{j=1}^d a_j\sin(2\pi x_j). \] A Fourier density is $\frac1{6i}\sum_j a_j[G_d(\xi-e_j)-G_d(\xi+e_j)]$. For either shift, $1+|\xi\pm e_j|\le2+|\xi|$, so its weighted integral is at most $2+1/\sqrt{2\pi}$ by Cauchy--Schwarz and the second moment. Hence the displayed density has weighted norm at most $(2+1/\sqrt{2\pi})/3<1$. Run the deterministic algorithm on the all-zero function and record its at most $m$ query points $x_i$. The $m\times d$ matrix $[\sin(2\pi(x_i)_j)]$ has a nonzero null vector; normalize it to $\|a\|_1=1$. Then $f_a$ and $-f_a$ both return zero at every query. Inductively their entire adaptive transcripts, and thus output $g$, equal the zero transcript. On the cube, $\phi_d\ge e^{-\pi/4}$. The coordinate sine functions are orthogonal and each has squared integral $1/2$. Since $1=\|a\|_1\le\sqrt d\|a\|_2$, \[ \|f_a\|_2\ge\frac{e^{-\pi/4}}{3\sqrt2}\|a\|_2 \ge\frac{e^{-\pi/4}}{3\sqrt{2d}}. \] Finally $2\|f_a\|_2\le\|f_a-g\|_2+\|-f_a-g\|_2$, so one of the indistinguishable inputs has at least the claimed error. ∎
This supplies a limited pre-asymptotic obstruction using the source's actual Fourier-density definition. The adversarial function depends on the deterministic zero transcript. It does not directly give a lower bound on expected error of randomized sampling, because one cannot choose a different fixed input after seeing each random seed.

Verification and remaining gaps

Numerical nullspaces for several $m<d$ designs were checked for zero sample residuals, $\ell_1$ normalization and the $\ell_2$ lower bound. Gaussian weighted-moment bounds and numerical cube integrals were checked separately. These do not prove optimal rates or novelty. The source already reports fixed-dimension asymptotics; no improvement of those results, no matching dimension-explicit upper bound, no variational-Barron equivalence and no optimal uniform-random-sampling theorem is claimed.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/21. Relevant passages read: 2024/21 pp.1213--1215, exact Fourier-density unit ball, fixed-dimension theorem and three sampling questions.

https://arxiv.org/abs/2208.07605. Read primary abstract defining Fourier-density Barron spaces and fixed-dimension sampling rates; no claim to improve that attributed theorem.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed its written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch025/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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