Volume 32, Issue 5
A New Function Space from Barron Class and Application to Neural Network Approximation

Commun. Comput. Phys., 32 (2022), pp. 1361-1400.

Published online: 2023-01

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• Abstract

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 $L^q$ error bound for the two-layer neural network with the Barron spectrum space as the target function space, the rate is dimension independent.

65N30, 65M12, 41A46, 35J25

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@Article{CiCP-32-1361, author = {Meng , Yan and Ming , Pingbing}, title = {A New Function Space from Barron Class and Application to Neural Network Approximation}, journal = {Communications in Computational Physics}, year = {2023}, volume = {32}, number = {5}, pages = {1361--1400}, abstract = {

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 $L^q$ error bound for the two-layer neural network with the Barron spectrum space as the target function space, the rate is dimension independent.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0151}, url = {http://global-sci.org/intro/article_detail/cicp/21367.html} }
TY - JOUR T1 - A New Function Space from Barron Class and Application to Neural Network Approximation AU - Meng , Yan AU - Ming , Pingbing JO - Communications in Computational Physics VL - 5 SP - 1361 EP - 1400 PY - 2023 DA - 2023/01 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2022-0151 UR - https://global-sci.org/intro/article_detail/cicp/21367.html KW - Fourier transform, Besov space, Sobolev space, radial function, neural network. AB -

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 $L^q$ error bound for the two-layer neural network with the Barron spectrum space as the target function space, the rate is dimension independent.

Yan Meng & Pingbing Ming. (2023). A New Function Space from Barron Class and Application to Neural Network Approximation. Communications in Computational Physics. 32 (5). 1361-1400. doi:10.4208/cicp.OA-2022-0151
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