We present some comparisons between the approximation rates relevant to linear approximators and the rates relevant to neural networks, i.e. nonlinear approximators represented by sets of parametrized functions corresponding to a type of computational unit. Our analysis uses the concept of variation of a function with respect to a set. The comparison is made in terms of Kolmogorov n-width for linear spaces and a proper nonlinear n-width for the nonlinear context represented by neural networks. The results of this paper contribute to the theoretical understanding of the superiority of neural networks with respect to linear approximators in complex tasks, as is confirmed by a wide variety of applications (recognition of handwritten characters and spoken numerals, approximate solution of functional optimization problems from control theory, etc.).

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