mlsanityData to 14 Oct 2026

Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks

arXiv 2307.15285 ↗First posted 28 Jul 2023stat.ML cs.LG cs.NAcited by 11

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Abstract (arXiv)

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in ℝd+1 can be approximated in the Hausdorff distance by a sum of n line segments. The second is to determine optimal approximation rates in the uniform norm for shallow ReLUk neural networks on their variation spaces. The first of these problems has been solved for dneq 2,3, but when d=2,3 a logarithmic gap between the best upper and lower bounds remains. We close this gap, which completes the solution in all dimensions. For the second problem, our techniques significantly improve upon existing approximation rates when k≥ 1, and enable uniform approximation of both the target function and its derivatives.

Key facts. Cited by 11 papers in the map (data through 14 Oct 2026). Builds on 11 of its 54 references that are in the map. First posted on arXiv 28 Jul 2023.

Builds on

11 of 54 references.

References · 11

  1. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
  2. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 311
  3. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 114
  4. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  5. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  6. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 51
  7. 2021Characterization of the Variation Spaces Corresponding to Shallow Neural Networkscited by 48
  8. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  9. 2023Optimal rates of approximation by shallow ReLUk neural networks and applications to nonparametric regressioncited by 15
  10. 2017Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functionscited by 8
  11. 2019Dimension independent bounds for general shallow networkscited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 4; 2025: 2; 2026: 5.

11 papers.

  1. 2024On the rates of convergence for learning with convolutional neural networkscited by 10
  2. 2024Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transformcited by 6
  3. 2024Approximation Bounds for Recurrent Neural Networks with Application to Regressioncited by 4
  4. 2025Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spherescited by 2
  5. 2026Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalizationcited by 2
  6. 2024Universality of reservoir systems with recurrent neural networkscited by 1
  7. 2025Configuration-Dependent Lower Bounds for Approximation by Shallow ReLUk Networks on the Spherecited by 1
  8. 2026Do Neural Networks Really Beat the Curse of Dimensionality? A Bit-Complexity Viewcited by 1
  9. 2026From Approximation Rates to Loss-Landscape Barrier Decay in Shallow ReLU Networkscited by 0
  10. 2026Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networkscited by 0
  11. 2026Minimax rates for learning spectral Barron functions by deep ReLU neural networkscited by 0