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The tldr is in the abstract: "In deep learning terminology, this amounts to saying that besides a negligible set, all functions that can be implemented by a deep network of polynomial size, require an exponential size if one wishes to implement (or approximate) them with a shallow network."

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That's a very interesting result (assuming it's correct).

It certainly agrees with intuition based on analogy to boolean circuits where, for example, the parity function requires exponential circuit size for shallow circuits but only linear size for deep circuits, but I haven't heard of a proof of this for NN's before.

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