Generalized Mercer Kernels and Reproducing Kernel Banach Spaces.
Xu, Yuesheng.
Generalized Mercer Kernels and Reproducing Kernel Banach Spaces. - 1st ed. - 1 online resource (134 pages) - Memoirs of the American Mathematical Society Series ; v.258 . - Memoirs of the American Mathematical Society Series .
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.
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.
9781470450779
Kernel functions.
Geometric function theory.
Banach spaces.
Functions of complex variables.
Support vector machines.
Electronic books.
QA353.K47 .X8 2019
515/.732
Generalized Mercer Kernels and Reproducing Kernel Banach Spaces. - 1st ed. - 1 online resource (134 pages) - Memoirs of the American Mathematical Society Series ; v.258 . - Memoirs of the American Mathematical Society Series .
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.
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.
9781470450779
Kernel functions.
Geometric function theory.
Banach spaces.
Functions of complex variables.
Support vector machines.
Electronic books.
QA353.K47 .X8 2019
515/.732