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We provide several new depth-based separation results for feed-forward neuralnetworks, proving that various types of simple and natural functions can bebetter approximated using deeper networks than shallower ones, even if theshallower networks are much larger. This includes indicators of balls andellipses; non-linear functions which are radial with respect to the $L_1$ norm;and smooth non-linear functions. We also show that these gaps can be observedexperimentally: Increasing the depth indeed allows better learning thanincreasing width, when training neural networks to learn an indicator of a unitball.

Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks