Before revision: The main success stories of deep learning, starting with ImageNet, depend on  convolutional networks, which on certain tasks perform significantly better than traditional shallow classifiers, such as support vector machines . 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: The main success stories of deep learning, starting with ImageNet, depend on convolutional networks, which on certain tasks perform significantly better than traditional shallow classifiers, such as support vector machines , and also better than deep fully connected networks; but what is so 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.