学习理论中的K函数

IF 2 2区 数学 Q1 MATHEMATICS Analysis and Applications Pub Date : 2020-05-01 DOI:10.1142/s0219530519500192
Bao-huai Sheng, Jianli Wang
{"title":"学习理论中的K函数","authors":"Bao-huai Sheng, Jianli Wang","doi":"10.1142/s0219530519500192","DOIUrl":null,"url":null,"abstract":"[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219530519500192","citationCount":"3","resultStr":"{\"title\":\"On the K-functional in learning theory\",\"authors\":\"Bao-huai Sheng, Jianli Wang\",\"doi\":\"10.1142/s0219530519500192\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.\",\"PeriodicalId\":55519,\"journal\":{\"name\":\"Analysis and Applications\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2020-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1142/s0219530519500192\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219530519500192\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219530519500192","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3

摘要

[公式:见正文]-泛函在学习理论文献中用于研究基于核的正则化方案中的近似误差。在本文中,我们用[公式:见文本]研究了[公式:看文本]空间中的近似误差和[公式:见文]-泛函。为此,我们从分数阶导数给出了再生核Hilbert空间(RKHS)的一个新观点,并将诱导积分算子的幂视为不同阶的分数阶导数。然后用傅立叶乘子定义了广义平移算子,并由此定义了广义光滑模。建立了光滑模量与[公式:见正文]-泛函之间的一些一般强等价关系。作为应用,明确地给出了单位球和单位球上这两个量族之间的一些强等价关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
On the K-functional in learning theory
[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
3.90
自引率
4.50%
发文量
29
审稿时长
>12 weeks
期刊介绍: Analysis and Applications publishes high quality mathematical papers that treat those parts of analysis which have direct or potential applications to the physical and biological sciences and engineering. Some of the topics from analysis include approximation theory, asymptotic analysis, calculus of variations, integral equations, integral transforms, ordinary and partial differential equations, delay differential equations, and perturbation methods. The primary aim of the journal is to encourage the development of new techniques and results in applied analysis.
期刊最新文献
On the strong solution for a diffuse interface model of non-Newtonian two-phase flows Distributed SGD in Overparameterized Linear Regression Interpolatory Taylor and Lidstone series Author index Volume 21 (2023) Convergence Analysis of Deep Residual Networks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1