Before revision:  Is there something special about deep convolutional networks that other learning machines do not possess ? Recent results in approximation theory have shown that there is an exponential advantage of deep convolutional-like networks in approximating functions with hierarchical locality in their compositional structure.
After revision:  Is there something special about deep convolutional networks that other learning machines do not possess ? Recent results in approximation theory have shown that there is an exponential advantage of deep convolutional networks with or without shared weights in approximating functions with hierarchical locality in their compositional structure.