A New Function Space from Barron Class and Application to Neural Network Approximation

We introduce a new function space, dubbed as the Barron spectrum space, which arises from the target function space for the neural network approximation. We give a Bernstein type sufficient condition for functions in this space, and clarify the embedding among the Barron spectrum space, the Bessel potential space, the Besov space and the Sobolev space. Moreover, the unexpected smoothness and the decaying behavior of the radial functions in the Barron spectrum space have been investigated. As an application, we prove a dimension explicit Lq error bound for the two-layer neural network with the Barron spectrum space as the target function space, the rate is dimension independent.