@Article{CiCP-28-1838,
author = {Li , YingzhouCheng , Xiuyuan and Lu , Jianfeng},
title = {Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks},
journal = {Communications in Computational Physics},
year = {2020},
volume = {28},
number = {5},
pages = {1838--1885},
abstract = {<p style="text-align: justify;">Deep networks, especially convolutional neural networks (CNNs), have
been successfully applied in various areas of machine learning as well as to challenging
problems in other scientific and engineering fields. This paper introduces Butterfly-net,
a low-complexity CNN with structured and sparse cross-channel connections, together
with a Butterfly initialization strategy for a family of networks. Theoretical analysis of
the approximation power of Butterfly-net to the Fourier representation of input data
shows that the error decays exponentially as the depth increases. Combining Butterfly-net with a fully connected neural network, a large class of problems are proved to
be well approximated with network complexity depending on the effective frequency
bandwidth instead of the input dimension. Regular CNN is covered as a special case
in our analysis. Numerical experiments validate the analytical results on the approximation of Fourier kernels and energy functionals of Poisson&#39;s equations. Moreover,
all experiments support that training from Butterfly initialization outperforms training
from random initialization. Also, adding the remaining cross-channel connections,
although significantly increases the parameter number, does not much improve the
post-training accuracy and is more sensitive to data distribution.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0214},
url = {http://global-sci.org/intro/article_detail/cicp/18398.html}
}