Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces

Yuwen Li; Guozhi Zhang

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arXiv · 2501.11275

Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces

Yuwen Li·Guozhi Zhang

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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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Yuwen Li, Guozhi Zhang. 2025-01-20. Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces. https://arxiv.org/abs/2501.11275

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