TY - JOUR

T1 - Approximation Spaces of Deep Neural Networks

AU - Gribonval, Rémi

AU - Kutyniok, Gitta

AU - Nielsen, Morten

AU - Voigtlaender, Felix

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Approximation spaces

KW - Besov spaces

KW - Deep neural networks

KW - Direct estimates

KW - Inverse estimates

KW - Piecewise polynomials

KW - ReLU activation function

KW - Sparsely connected networks

UR - http://www.scopus.com/inward/record.url?scp=85105443182&partnerID=8YFLogxK

U2 - 10.1007/s00365-021-09543-4

DO - 10.1007/s00365-021-09543-4

M3 - Journal article

VL - 55

SP - 259

EP - 367

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 1

ER -