mlsanityData to 7 Oct 2026

Limitations on approximation by deep and shallow neural networks

arXiv 2212.02223 ↗First posted 30 Nov 2022stat.ML cs.LG math.FAcited by 4

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Abstract (arXiv)

We prove Carl's type inequalities for the error of approximation of compact sets K by deep and shallow neural networks. This in turn gives lower bounds on how well we can approximate the functions in K when requiring the approximants to come from outputs of such networks. Our results are obtained as a byproduct of the study of the recently introduced Lipschitz widths.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 10 of its 33 references that are in the map. First posted on arXiv 30 Nov 2022.

Builds on

10 of 33 references.

References · 10

  1. 2016Error bounds for approximations with deep ReLU networkscited by 567
  2. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 304
  3. 2016Benefits of depth in neural networkscited by 248
  4. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  5. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 165
  6. 2020Deep Network Approximation for Smooth Functionscited by 163
  7. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 95
  8. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  9. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
  10. 2022A general approximation lower bound in Lp norm, with applications to feed-forward neural networkscited by 12

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Cited by year

Citing papers per year: 2023: 1; 2024: 1; 2025: 2.

4 papers.

  1. 2025Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapterscited by 9
  2. 2024Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machinescited by 2
  3. 2025Are all models wrong? Fundamental limits in distribution-free empirical model falsificationcited by 1
  4. 2023Neural networks: deep, shallow, or in between?cited by 0