Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units

doi:10.4208/cicp.OA-2019-0168

CSpace

	Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units
	Li, Bo 1,2,3; Tang, Shanshan 4; Yu, Haijun 1,2,3
	2020-02-01
发表期刊	COMMUNICATIONS IN COMPUTATIONAL PHYSICS
ISSN	1815-2406
卷号	27 期号:2 页码:379-411
摘要	Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance epsilon, by our constructive proofs, are in general O(log1/epsilon) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.
关键词	Deep neural network high dimensional approximation sparse grids rectified linear unit rectified power unit rectified quadratic unit
DOI	10.4208/cicp.OA-2019-0168
收录类别	SCI
语种	英语
资助项目	China National Program on Key Basic Research Project[2015CB856003] ; NNSFC[11771439] ; NNSFC[91852116] ; China Science Challenge Project[TZ2018001]
WOS研究方向	Physics
WOS类目	Physics, Mathematical
WOS记录号	WOS:000501534800002
出版者	GLOBAL SCIENCE PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/50326
专题	中国科学院数学与系统科学研究院
通讯作者	Yu, Haijun
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, NCMIS, Beijing 100190, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, LSEC, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 4.China Justice Big Data Inst, Beijing 100043, Peoples R China
推荐引用方式 GB/T 7714	Li, Bo,Tang, Shanshan,Yu, Haijun. Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units[J]. COMMUNICATIONS IN COMPUTATIONAL PHYSICS,2020,27(2):379-411.
APA	Li, Bo,Tang, Shanshan,&Yu, Haijun.(2020).Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units.COMMUNICATIONS IN COMPUTATIONAL PHYSICS,27(2),379-411.
MLA	Li, Bo,et al."Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units".COMMUNICATIONS IN COMPUTATIONAL PHYSICS 27.2(2020):379-411.