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We define a notion of complexity, which quantifies the nonlinearity of thecomputation of a neural network, as well as a complementary measure of theeffective dimension of feature representations. We investigate theseobservables both for trained networks for various datasets as well as exploretheir dynamics during training, uncovering in particular power law scaling.These observables can be understood in a dual way as uncovering hidden internalstructure of the datasets themselves as a function of scale or depth. Theentropic character of the proposed notion of complexity should allow totransfer modes of analysis from neuroscience and statistical physics to thedomain of artificial neural networks. The introduced observables can be appliedwithout any change to the analysis of biological neuronal systems.

Complexity for deep neural networks and other characteristics of deep feature representations