Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces

Yuwen Li, Guozhi Zhang
1/20/2025
cs.LGmath.NA

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Abstract

This paper investigates the $L_p$ approximation error for higher order Korobov functions using deep convolutional neural networks (CNNs) with ReLU activation. For target functions having a mixed derivative of order m+1 in each direction, we improve classical approximation rate of second order to (m+1)-th order (modulo a logarithmic factor) in terms of the depth of CNNs. The key ingredient in our analysis is approximate representation of high-order sparse grid basis functions by CNNs. The results suggest that higher order expressivity of CNNs does not severely suffer from the curse of dimensionality.

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Cite this paper

@article{li2025higher,
  title  = {Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces},
  author = {Yuwen Li and Guozhi Zhang},
  year   = {2025},
  eprint = {2501.11275},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2501.11275v1}
}

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