Approximation Spaces of Deep Neural Networks

Rémi Gribonval, Gitta Kutyniok*, Morten Nielsen, Felix Voigtlaender

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

27 Citations (Scopus)


We study the expressivity of deep neural networks. Measuring a network’s complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be endowed with a (quasi)-norm that makes it a linear function space, called approximation space. We establish that allowing the networks to have certain types of “skip connections” does not change the resulting approximation spaces. We also discuss the role of the network’s nonlinearity (also known as activation function) on the resulting spaces, as well as the role of depth. For the popular ReLU nonlinearity and its powers, we relate the newly constructed spaces to classical Besov spaces. The established embeddings highlight that some functions of very low Besov smoothness can nevertheless be well approximated by neural networks, if these networks are sufficiently deep.
Original languageEnglish
JournalConstructive Approximation
Issue number1
Pages (from-to)259-367
Publication statusPublished - 2022


  • Approximation spaces
  • Besov spaces
  • Deep neural networks
  • Direct estimates
  • Inverse estimates
  • Piecewise polynomials
  • ReLU activation function
  • Sparsely connected networks


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