A New Function Space from Barron Class and Application to Neural Network Approximation (Communications in Computational Physics, 2023)
We introduce a new function space, dubbed as the Barron spectrum space,which arises from the target function space for the neural network approximation. Wegive a Bernstein type sufficient condition for functions in this space, and clarify theembedding among the Barron spectrum space, the Bessel potential space, the Besovspace and the Sobolev space. Moreover, the unexpected smoothness and the decayingbehavior of the radial functions in the Barron spectrum space have been investigated.As an application, we prove a dimension explicit $L^q$ error bound for the two-layerneural network with the Barron spectrum space as the target function space, the rateis dimension independent.