Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of   Dimensionality: a Review
  The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage. 
Tomaso Poggio, Hrushikesh Mhaskar, Lorenzo Rosasco, Brando Miranda,   Qianli Liao
cs.LG