On sharpness of error bounds for multivariate neural network approximation

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Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best non-linear approximation by such sums of ridge functions. Error bounds are presented in terms of moduli of smoothness. The main focus, however, is to prove that the bounds are best possible. To this end, counterexamples are constructed with a non-linear, quantitative extension of the uniform boundedness principle. They show sharpness with respect to Lipschitz classes for the logistic activation function and for certain piecewise polynomial activation functions. The paper is based on univariate results in Goebbels (Res Math 75(3):1–35, 2020. https://rdcu.be/b5mKH)