Deep vs. shallow networks : An approximation theory perspective
This work addresses a foundational problem in machine learning theory for researchers, but it is incremental as it builds on existing results.
The paper investigates why deep convolutional neural networks outperform shallow ones in function approximation, proposing a new definition of relative dimension to explain how deep networks exploit sparsity to reduce complexity.
The paper briefy reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation problems than shallow, one-hidden layer architectures. The paper announces new results for a non-smooth activation function - the ReLU function - used in present-day neural networks, as well as for the Gaussian networks. We propose a new definition of relative dimension to encapsulate different notions of sparsity of a function class that can possibly be exploited by deep networks but not by shallow ones to drastically reduce the complexity required for approximation and learning.