Regularization in Statistics, 3.4 Bayes and regularization It is asserted, with some justification, that Bayesian methods regularize automatically

https://normaldeviate.wordpress.com/2013/01/19/bootstrapping-and-subsampling-part-i/

Integral Equation Analysis of EM Scattering from ...

www.hindawi.com/journals/ijap/2015/274307/

by J Su - ‎2015 - ‎Related articles

Jul 8, 2015 - ... implementation are independent of integral equation kernel (Green's function). Therefore, the ACA is very well suited for accelerating integral ...

3.4 Bayes and regularization It is asserted, with some justification, that Bayesian methods regularize automatically

https://www.stat.berkeley.edu/~bickel/Test_BickelLi.pdf

Regularization in Statistics Peter J. Bickel Department of Statistics University of California, Berkeley, USA Bo Li School of Economics and Management Tsinghua University, China Abstract This paper is a selective review of the regularization methods scattered in statistics literature. We introduce a general conceptual approach to regularization and fit most existing methods into it. We have tried to focus on the importance of regularization when dealing with today’s high-dimensional objects: data and models. A wide range of examples are discussed, including nonparametric regression, boosting, covariance matrix estimation, principal component estimation, subsampling. Key Words: Regularization, linear regression, nonparametric regression, boosting, covariance matrix, principal component, bootstrap, subsampling, model selection. AMS subject classification: Primary 62G08, 62H12; Secondary 62F12, 62G20, 62H25. 1 Introduction The concept of regularization was first introduced in the context of solving integral equation numerically by Tikhonov (1943). As is well known if f ∈ L2(R) and K(x,y) is a smooth kernel, the range of the operator A, R(A), A : L2(R) 7→ L2(R) with (Af)(y) ≡ R K(x,y)f(x)dx is dense in L2(R) but not onto. Thus, the inverse A−1 is ill-posed. The solution to the equation Af = g (1.1) is hard to determine since approximations to g easily lie outside R(A). Tikhonov’s solution was to replace (1.1) by the minimization of kAf − gk 2+γW(f), where the Tikhonov factor γ > 0 is a regularization parameter and W(f) is a smoothness penalty such as R [f 0 (x)]2dx. Numerical (finite ∗Correspondence to: Peter J. Bickel. Department of Statis

• Kernel of integral equation
  积分方程的核
• For the given sample points , and matrix formed by covariance function with sample points as parameters , when the number of sample points approaches infinite , it is proven that this matrix spectrum will approach the spectral approach theorem for positive - definite kernel of integral equation
  对给定的样本点，由样本点为变量的协方差函数构成的矩阵，当样本点个数趋于无穷大时，证明此矩阵谱逼近于积分方程正定核的谱逼近定理