Kernel-Induced Sampling Theorem for Translation-Invariant Reproducing Kernel Hilbert Spaces with Uniform Sampling

2018 ◽  
Author(s):  
Akira Tanaka
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Translation Invariant ◽  
Uniform Sampling ◽  
Kernel Hilbert Spaces

scholarly journals ON THE INCLUSION RELATION OF REPRODUCING KERNEL HILBERT SPACES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350014 ◽  
Author(s):  
HAIZHANG ZHANG ◽  
LIANG ZHAO
Keyword(s):  
Applied Sciences ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Inclusion Relation ◽  
Feature Maps ◽  
Translation Invariant ◽  
Inclusion Relations ◽  
Kernel Hilbert Spaces

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

scholarly journals VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY

2010 ◽  
Vol 08 (01) ◽  
pp. 19-61 ◽  
Author(s):  
C. CARMELI ◽  
E. DE VITO ◽  
A. TOIGO ◽  
V. UMANITÀ
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Abelian Groups ◽  
Reproducing Kernel Hilbert Spaces ◽  
Feature Maps ◽  
Translation Invariant ◽  
Kernel Hilbert Spaces ◽  
Vector Valued ◽  
Invariant Kernels

This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular, we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.

scholarly journals Aspects of prediction

2014 ◽  
Vol 51 (A) ◽  
pp. 189-201
Author(s):  
N. H. Bingham ◽  
Badr Missaoui
Keyword(s):  
Hilbert Spaces ◽  
Continuous Time ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Stationary Processes ◽  
Prediction Theory ◽  
Volatility Clustering ◽  
Infinite Dimensional ◽  
Kernel Hilbert Spaces

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

scholarly journals Aspects of prediction

2014 ◽  
Vol 51 (A) ◽  
pp. 189-201 ◽  
Author(s):  
N. H. Bingham ◽  
Badr Missaoui
Keyword(s):  
Hilbert Spaces ◽  
Continuous Time ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Stationary Processes ◽  
Prediction Theory ◽  
Volatility Clustering ◽  
Infinite Dimensional ◽  
Kernel Hilbert Spaces

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

INDEFINITE KERNEL NETWORK WITH DEPENDENT SAMPLING

2013 ◽  
Vol 11 (05) ◽  
pp. 1350020 ◽  
Author(s):  
HONGWEI SUN ◽  
QIANG WU
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Function Class ◽  
Positive Definiteness ◽  
Reproducing Kernel Hilbert Spaces ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Indefinite Kernel ◽  
Learning Rates ◽  
Kernel Hilbert Spaces

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

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