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We consider a family of deep neural networks consisting of two groups ofconvolutional layers, a downsampling operator, and a fully connected layer. Thenetwork structure depends on two structural parameters which determine thenumbers of convolutional layers and the width of the fully connected layer. Weestablish an approximation theory with explicit approximation rates when theapproximated function takes a composite form $f\circ Q$ with a featurepolynomial $Q$ and a univariate function $f$. In particular, we prove that sucha network can outperform fully connected shallow networks in approximatingradial functions with $Q(x) =|x|^2$, when the dimension $d$ of data from$\mathbb{R}^d$ is large. This gives the first rigorous proof for thesuperiority of deep convolutional neural networks in approximating functionswith special structures. Then we carry out generalization analysis forempirical risk minimization with such a deep network in a regression frameworkwith the regression function of the form $f\circ Q$. Our network structurewhich does not use any composite information or the functions $Q$ and $f$ canautomatically extract features and make use of the composite nature of theregression function via tuning the structural parameters. Our analysis providesan error bound which decreases with the network depth to a minimum and thenincreases, verifying theoretically a trade-off phenomenon observed for networkdepths in many practical applications.

Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions