mlsanityData to 14 Oct 2026

Time-Frequency Analysis for Neural Networks

arXiv 2512.15992 ↗First posted 17 Dec 2025math.NA cs.IT cs.LGcited by 2

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Abstract (arXiv)

We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces Mp,qm(Rd), we prove dimension-independent approximation rates in Sobolev norms Wn,r(Omega) for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for f ∈ Mp,qm(Rd) one can achieve [ |f - fN|_Wn,r(Omega) lesssim N-1/2,|f|_Mp,qm(Rd), ] on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on Rd using weighted modulation dictionaries, and derive consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.

Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 20 of its 58 references that are in the map. First posted on arXiv 17 Dec 2025.

Builds on

20 of 58 references.

References · 20

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8 more
  1. 2017Deep Relaxation: partial differential equations for optimizing deep neural networkscited by 27
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  8. 2024Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domainscited by 5

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0
  2. 2026Approximation Rates for Metaplectic Neural Networkscited by 0