Function approximation by deep networks

TitleFunction approximation by deep networks
Publication TypeJournal Article
Year of Publication2020
AuthorsMhaskar, HN, Poggio, T
JournalCommunications on Pure & Applied Analysis
Pagination4085 - 4095
Date Published08/2020
Keywordsapproximation on the Euclidean sphere, deep networks, degree of approximation

We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thus, the blessing of compositionality mitigates the curse of dimensionality. On the other hand, a theorem called good propagation of errors allows to "lift" theorems about shallow networks to those about deep networks with an appropriate choice of norms, smoothness, etc. We illustrate this in three contexts where each channel in the deep network calculates a spherical polynomial, a non-smooth ReLU network, or another zonal function network related closely with the ReLU network.


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