scholarly journals Reproducing Kernel Banach Spaces with the ℓ1 Norm II: Error Analysis for Regularized Least Square Regression

2011 ◽  
Vol 23 (10) ◽  
pp. 2713-2729 ◽  
Author(s):  
Guohui Song ◽  
Haizhang Zhang
Keyword(s):  
Banach Spaces ◽  
Reproducing Kernel ◽  
Sampling Error ◽  
Approximation Error ◽  
Learning Rate ◽  
Least Square ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Regularization Error ◽  
Reproducing Kernel Banach Spaces

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error, and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the ℓ1 norm can be applied to improve the learning rate estimate of ℓ1-regularization in machine learning.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

CHARACTERIZATION OF IMAGE SPACE OF A WAVELET TRANSFORM

2006 ◽  
Vol 04 (03) ◽  
pp. 547-557 ◽  
Author(s):  
CAIXIA DENG ◽  
YULING QU ◽  
LIJUAN GU
Keyword(s):  
Wavelet Transform ◽  
Truncation Error ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Image Space ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Scope Of Application

In this paper, Journe wavelet function is introduced as a wavelet generating function. The expression of reproducing kernel function for the image space of this wavelet transform is obtained based on the fact that the image space of the wavelet transform is a reproducing kernel Hilbert space. Then the isometric identity of Journe wavelet transform is obtained. The connections between the image space of the wavelet transform and the image space of the known reproducing kernel space are established by the theories of reproducing kernel. The properties and the structures of the image space of the wavelet transform can be characterized by the properties and the structures of the image space of the known reproducing kernel space. Using the ideas of reproducing kernel, we consider there are relations between the wavelet transform and the sampling theorem. Meanwhile, the approximations in sampling theorems is shown and the truncation error is given. This provides a theoretical basis for us to study the image space of the general wavelet transform and broadens the scope of application of theories of the reproducing kernel space.

scholarly journals A New Method Based on the RKHSM for Solving Systems of Nonlinear IDDEs with Proportional Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Changbo He ◽  
Xueqin Lv ◽  
Jing Niu
Keyword(s):  
Iterative Method ◽  
Delay Differential Equations ◽  
Reproducing Kernel ◽  
Infinite Delay ◽  
Computational Method ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Proportional Delays ◽  
Computation Efficiency ◽  
Delay Differential

An efficient computational method is given in order to solve the systems of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays. Representation of the solution and an iterative method are established in the reproducing kernel space. Some examples are displayed to demonstrate the computation efficiency of the method.

scholarly journals A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Er Gao ◽  
Songhe Song ◽  
Xinjian Zhang
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Space Method ◽  
Term Approximation

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional orderq∈(1,2]based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so then-term approximation. At the same time, then-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

scholarly journals Frames for operators in Banach spaces via semi-inner products

2016 ◽  
Vol 14 (03) ◽  
pp. 1650011 ◽  
Author(s):  
Bahram Dastourian ◽  
Mohammad Janfada
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Reproducing Kernel ◽  
Atomic System ◽  
Inner Product ◽  
Reconstruction Formula ◽  
Frame Operator ◽  
Atomic Systems ◽  
Perturbation Result ◽  
Reproducing Kernel Banach Spaces

In this paper, the concept of a family of local atoms in a Banach space is introduced by using a semi-inner product (s.i.p.). Then this concept is generalized to an atomic system for operators in Banach spaces. We also give some characterizations of atomic systems leading to new frames for operators. In addition, a reconstruction formula is obtained. The characterizations of atomic systems allow us to state some results for sampling theory in s.i.p reproducing kernel Banach spaces. Finally, we define the concept of frame operator for these kinds of frames in Banach spaces and then we establish a perturbation result in this framework.

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