On Sharpness of Error Bounds for Multivariate Neural Network Approximation

Single hidden layer feedforward neural networks can represent multivariate\nfunctions that are sums of ridge functions. These ridge functions are defined\nvia an activation function and customizable weights. The paper deals with best\nnon-linear approximation by such sums of ridge functions. Error bounds are\npresented in terms of moduli of smoothness. The main focus, however, is to\nprove that the bounds are best possible. To this end, counterexamples are\nconstructed with a non-linear, quantitative extension of the uniform\nboundedness principle. They show sharpness with respect to Lipschitz classes\nfor the logistic activation function and for certain piecewise polynomial\nactivation functions. The paper is based on univariate results in (Goebbels,\nSt.: On sharpness of error bounds for univariate approximation by single hidden\nlayer feedforward neural networks. Results Math 75 (3), 2020, article 109,\nhttps://rdcu.be/b5mKH).\n