mlsanityData to 14 Oct 2026

Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

arXiv 2605.31152 ↗First posted 29 May 2026stat.ML cs.LG cs.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in Lp([0,1]d) norm and the approximator is a network with width W and depth L, recent works have proven the supper approximation rate O((WL)-2s/d) for Besov space Bsq,r([0,1]d) under the Sobolev embedding condition s/d>1/q-1/p. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate O((WL)^-2tildes) for anisotropic Besov space B^boldsymbolsq,r([0,1]d) with anisotropic smoothness boldsymbols=(s1,dots,sd) under the embedding condition tildes > 1/q-1/p, where the mean smoothness tildes = (sumi=1d si-1)-1. For mixed smooth Besov space MBsq,r([0,1]d) with mixed smoothness s>1/q-1/p, we show that the approximation rate O((WL)-2s) holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 26 of its 64 references that are in the map. First posted on arXiv 29 May 2026.

Builds on

26 of 64 references.

References · 26

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  2. 2016Error bounds for approximations with deep ReLU networkscited by 574
  3. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 433
  4. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 311
  5. 2018Universality of Deep Convolutional Neural Networkscited by 176
  6. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  7. 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 165
  8. 2020Deep Network Approximation for Smooth Functionscited by 164
  9. 2018Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionalitycited by 162
  10. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  11. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 96
  12. 2019The phase diagram of approximation rates for deep neural networkscited by 88
14 more
  1. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  2. 2019Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recoverycited by 73
  3. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 51
  4. 2018Fast convergence rates of deep neural networks for classificationcited by 46
  5. 2019Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov spacecited by 46
  6. 2021Near-Minimax Optimal Estimation With Shallow ReLU Neural Networkscited by 41
  7. 2019Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networkscited by 40
  8. 2023Nonparametric regression using over-parameterized shallow ReLU neural networkscited by 24
  9. 2022Approximation bounds for norm constrained neural networks with applications to regression and GANscited by 22
  10. 2024On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networkscited by 13
  11. 2022A general approximation lower bound in Lp norm, with applications to feed-forward neural networkscited by 12
  12. 2024On the rates of convergence for learning with convolutional neural networkscited by 10
  13. 2025Higher Order Approximation Rates for ReLU CNNs in Korobov Spacescited by 4
  14. 2013Learning Tensors in Reproducing Kernel Hilbert Spaces with Multilinear Spectral Penaltiescited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Widthcited by 0
  2. 2026Shallow neural network approximation in mixed Sobolev spacescited by 0
  3. 2026Minimax rates for learning spectral Barron functions by deep ReLU neural networkscited by 0