On Reproducing Kernel Banach Spaces:Generic Definitions and Unified Framework of Constructions  

作　　者：Rong Rong LIN Hai Zhang ZHANG Jun ZHANG 

机构地区：[1]School of Mathematics and Statistics,Guangdong University of Technology,Guangzhou 510520,P.R.China [2]School of Mathematics(Zhuhai),and Guangdong Province Key Laboratory of Computational Science,Sun Yat-sen University,Zhuhai 519082,P.R.China [3]Department of Psychology and Department of Statistics,University of Michigan,Ann Arbor,MI48109,USA

出　　处：《Acta Mathematica Sinica,English Series》2022年第8期1459-1483,共25页数学学报（英文版）

基　　金：Supported by Natural Science Foundation of China(Grant Nos.11971490,11901595);Natural Science Foundation of Guangdong Province(Grant No.2018A030313841);Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(Grant No.2020B1212060032);AFOSR(Grant No.FA9550-19-1-0213)through a subcontract from University of California,Los Angeles。

摘　　要：Recently,there has been emerging interest in constructing reproducing kernel Banach spaces(RKBS)for applied and theoretical purposes such as machine learning,sampling reconstruction,sparse approximation and functional analysis.Existing constructions include the reflexive RKBS via a bilinear form,the semi-inner-product RKBS,the RKBS with?~1 norm,the p-norm RKBS via generalized Mercer kernels,etc.The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction.Moreover,relations among those constructions are unclear.We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction.Furthermore,we propose a framework of constructing RKBSs that leads to new RKBSs based on Orlicz spaces and unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps.Finally,we develop representer theorems for machine learning in RKBSs constructed in our framework,which also unifies representer theorems in existing RKBSs.

关 键 词：Reproducing kernel Banach spaces feature maps reproducing kernels machine learning the representer theorem 

分 类 号：O177.2[理学—数学]