Nonlocal techniques for the analysis of deep ReLU neural network approximations
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Abstract (arXiv)
Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a system of piece-wise linear functions, which can be easily reproduced by artificial neural networks with the ReLU activation function and which form a Riesz basis of L2([0,1]). This work was generalized by two of the authors to the multivariate setting. We show that this system serves as a Riesz basis also for Sobolev spaces Ws([0,1]d) and Barron classes mathbb Bs([0,1]d) with smoothness 0<s<1. We apply this fact to re-prove some recent results on the approximation of functions from these classes by deep neural networks. Our proof method avoids using local approximations and allows us to track also the implicit constants as well as to show that we can avoid the curse of dimension. Moreover, we also study how well one can approximate Sobolev and Barron functions by ANNs if only function values are known.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 21 of its 69 references that are in the map. First posted on arXiv 7 Apr 2025.
Builds on
21 of 69 references.
References · 21
- 2016Error bounds for approximations with deep ReLU networkscited by 574
- 2016On the Expressive Power of Deep Neural Networkscited by 306
- 2016Benefits of depth in neural networkscited by 250
- 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
- 2020Deep Network Approximation for Smooth Functionscited by 164
- 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 96
- 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
- 2019Deep Neural Network Approximation Theorycited by 93
- 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 91
- 2015Representation Benefits of Deep Feedforward Networkscited by 86
- 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
- 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
9 more
- 2016Deep vs. shallow networks : An approximation theory perspectivecited by 63
- 2020Representation formulas and pointwise properties for Barron functionscited by 62
- 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 51
- 2020On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamicscited by 39
- 2020Neural network approximation and estimation of classifiers with classification boundary in a Barron classcited by 28
- 2016Learning Functions: When Is Deep Better Than Shallowcited by 19
- 2018A comparison of deep networks with ReLU activation function and linear spline-type methodscited by 18
- 2022Lp sampling numbers for the Fourier-analytic Barron spacecited by 6
- 2020A priori estimates for classification problems using neural networkscited by 5
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Citing papers per year: 2025: 2.
2 papers.