Deep ReLU networks approximate anisotropic Besov functions at rate O((WL)^(-2\tilde{s})) and mixed-smooth Besov functions at rate O((WL)^(-2s)) up to logs, with matching lower bounds up to logs.
Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces
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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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Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks
Deep ReLU networks approximate anisotropic Besov functions at rate O((WL)^(-2\tilde{s})) and mixed-smooth Besov functions at rate O((WL)^(-2s)) up to logs, with matching lower bounds up to logs.