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Classical results in neural network approximation theory show how arbitrarycontinuous functions can be approximated by networks with a single hiddenlayer, under mild assumptions on the activation function. However, theclassical theory does not give a constructive means to generate the networkparameters that achieve a desired accuracy. Recent results have demonstratedthat for specialized activation functions, such as ReLUs and some classes ofanalytic functions, high accuracy can be achieved via linear combinations ofrandomly initialized activations. These recent works utilize specializedintegral representations of target functions that depend on the specificactivation functions used. This paper defines mollified integralrepresentations, which provide a means to form integral representations oftarget functions using activations for which no direct integral representationis currently known. The new construction enables approximation guarantees forrandomly initialized networks for a variety of widely used activationfunctions.

Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control