Generalized Mercer Kernels and Reproducing Kernel Banach Spaces
This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implem...
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| Hlavní autori: | , |
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| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, Rhode Island
American Mathematical Society
2019
|
| Vydanie: | 1 |
| Edícia: | Memoirs of the American Mathematical Society |
| Predmet: | |
| ISBN: | 9781470435509, 1470435500 |
| ISSN: | 0065-9266, 1947-6221 |
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| Abstract | This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing
kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be
well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity,
separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and
universal approximation. Then, we develop a new concept of generalized Mercer kernels to construct |
|---|---|
| AbstractList | This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First the authors verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, they develop a new concept of generalized Mercer kernels to construct $p$-norm RKBSs for $1\leq p\leq\infty$. This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, we develop a new concept of generalized Mercer kernels to construct |
| Author | Xu, Yuesheng Ye, Qi |
| Author_xml | – sequence: 1 fullname: Xu, Yuesheng – sequence: 2 fullname: Ye, Qi |
| BackLink | https://cir.nii.ac.jp/crid/1130000795945869568$$DView record in CiNii |
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| Copyright | Copyright 2019 American Mathematical Society |
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| DOI | 10.1090/memo/1243 |
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| Keywords | Generalized Mercer Kernels Machine Learning Support Vector Machines Sparse Learning Methods Positive Definite Kernels Reproducing Kernel Banach Spaces |
| LCCN | 2019013169 |
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| Notes | March 2019, volume 258, number 1243 (seventh of 7 numbers) Includes bibliographical references and index |
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| Snippet | This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing
kernel Hilbert spaces... This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces... |
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| SubjectTerms | Banach spaces Functions of complex variables Geometric function theory Kernel functions Support vector machines |
| TableOfContents | Introduction
--
Reproducing Kernel Banach Spaces
--
Generalized Mercer Kernels
--
Positive Definite Kernels
--
Support Vector Machines
--
Concluding Remarks
--
Acknowledgments Cover -- Title page -- Chapter 1. Introduction -- 1.1. Machine Learning in Banach Spaces -- 1.2. Overview of Kernel-based Function Spaces -- 1.3. Main Results -- Chapter 2. Reproducing Kernel Banach Spaces -- 2.1. Reproducing Kernels and Reproducing Kernel Banach Spaces -- 2.2. Density, Continuity, and Separability -- 2.3. Implicit Representation -- 2.4. Imbedding -- 2.5. Compactness -- 2.6. Representer Theorem -- 2.7. Oracle Inequality -- 2.8. Universal Approximation -- Chapter 3. Generalized Mercer Kernels -- 3.1. Constructing Generalized Mercer Kernels -- 3.2. Constructing -norm Reproducing Kernel Banach Spaces for 1< -- < -- ∞ -- 3.3. Constructing 1-norm Reproducing Kernel Banach Spaces -- Chapter 4. Positive Definite Kernels -- 4.1. Definition of Positive Definite Kernels -- 4.2. Constructing Reproducing Kernel Banach Spaces by Eigenvalues and Eigenfunctions -- 4.3. Min Kernels -- 4.4. Gaussian Kernels -- 4.5. Power Series Kernels -- Chapter 5. Support Vector Machines -- 5.1. Background of Support Vector Machines -- 5.2. Support Vector Machines in -norm Reproducing Kernel Banach Spaces for 1< -- < -- ∞ -- 5.3. Support Vector Machines in 1-norm Reproducing Kernel Banach Spaces -- 5.4. Sparse Sampling in 1-norm Reproducing Kernel Banach Spaces -- 5.5. Support Vector Machines in Special Reproducing Kernel Banach Spaces -- Chapter 6. Concluding Remarks -- Acknowledgments -- Bibliography -- Index -- Back Cover |
| Title | Generalized Mercer Kernels and Reproducing Kernel Banach Spaces |
| URI | https://www.ams.org/memo/1243/ https://cir.nii.ac.jp/crid/1130000795945869568 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=5770276 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781470450779 |
| Volume | 258 |
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