Deep ReLU Networks Preserve Expected Length
This provides theoretical insights into the complexity and generalization of deep neural networks, addressing a foundational issue in machine learning.
The paper tackles the problem of understanding how deep ReLU networks distort input curves, challenging the belief that length grows exponentially with depth. It proves that expected length distortion does not increase with depth and shrinks slightly under standard random initialization, supported by experiments.
Assessing the complexity of functions computed by a neural network helps us understand how the network will learn and generalize. One natural measure of complexity is how the network distorts length - if the network takes a unit-length curve as input, what is the length of the resulting curve of outputs? It has been widely believed that this length grows exponentially in network depth. We prove that in fact this is not the case: the expected length distortion does not grow with depth, and indeed shrinks slightly, for ReLU networks with standard random initialization. We also generalize this result by proving upper bounds both for higher moments of the length distortion and for the distortion of higher-dimensional volumes. These theoretical results are corroborated by our experiments.