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Deep neural network approximation of composite functions without the curse of dimensionality

arXiv 2304.05790 ↗First posted 12 Apr 2023math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) activation without the curse of dimensionality. In other words, the number of DNN parameters grows at most polynomially in the input dimension and the approximation error. The functions in our class can be expressed as a potentially unbounded number of compositions of special functions which include products, maxima, and certain parallelized Lipschitz continuous functions.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 17 of its 36 references that are in the map. First posted on arXiv 12 Apr 2023.

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References · 17

  1. 2016Error bounds for approximations with deep ReLU networkscited by 574
  2. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 320
  3. 2015The Power of Depth for Feedforward Neural Networkscited by 183
  4. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  5. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  6. 2020Deep Network Approximation for Smooth Functionscited by 164
  7. 2016Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Reviewcited by 155
  8. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  9. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  10. 2016Deep vs. shallow networks : An approximation theory perspectivecited by 63
  11. 2016Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networkscited by 59
  12. 2017Depth Separation for Neural Networkscited by 46
5 more
  1. 2020Expressivity of Deep Neural Networkscited by 32
  2. 2016Learning Functions: When Is Deep Better Than Shallowcited by 19
  3. 2019Full error analysis for the training of deep neural networkscited by 17
  4. 2020Strong overall error analysis for the training of artificial neural networks via random initializationscited by 4
  5. 2023The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionalitycited by 1

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