A random net is a shallow neural network where the hidden layer is frozen with random assignment and the output layer is trained by convex optimization. Using random weights for a hidden layer is an effective method to avoid the inevitable non-convexity in standard gradient descent learning. It has recently been adopted in the study of deep learning theory. Here, we investigate the expressive power of random nets. We show that, despite the well-known fact that a shallow neural network is a universal approximator, a random net cannot achieve zero approximation error even for smooth functions. In particular, we prove that for a class of smooth functions, if the proposal distribution is compactly supported, then a lower bound is positive. Based on the ridgelet analysis and harmonic analysis for neural networks, the proof uses the Plancherel theorem and an estimate for the truncated tail of the parameter distribution. We corroborate our theoretical results with various simulation studies, and generally two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.
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