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Limitations of approximation capabilities of shallow perceptron networks are investigated. Lower bounds on approximation errors are derived for binary-valued functions on finite domains. It is proven that unless the number of network units is sufficiently large (larger than any polynomial of the logarithm of the size of the domain) a good approximation cannot be achieved for almost any uniformly randomly...
Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable {−1,1}-valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabilistic...
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