Error Bound And Learning Rate Of Kernel-based Principal Component Learning Algorithm

Principal component analysis（PCA）is a commonly used dimensionality reduction method that achieves dimensionality reduction with minimal loss of information from the original data.PCA finds the direction with the largest variance,and the projection along this direction is called the principal component.Kernel principal component analysis（KPCA）is a dimensionality reduction method used to process nonlinear input data.Although the mathematical theoretical research on the algorithm of kernel principal component analysis is a relatively new topic,the algorithm of another commonly used dimensionality reduction method,kernel canonical correlation analysis（KCCA）,has already been researched.By covariance operator,Fukumizu^[1]et al.constructed error analysis theory of kernel canonical correlation analysis algorithm.In this paper,we will construct mathematical framework of kernel principal component analysis algorithm using the covariance operator as the main tool,for the uncentered case,we will obtain the error bound and learning rate of the learning algorithm of kernel-based principal component analysis.In this paper,we mainly focus on the learning algorithm of kernel principal component analysis.By analyzing the principle of kernel principal component analysis,it can be seen that its direction and projection are in one-to-one correspondence,which allows us to transform it into a function learning problem.Next the learning algorithm of kernel-based principal component analysis is given using the covariance operator.For the uncentered case,in order to calculate the learning error of the algorithm,we introduce the distance from the vector to the feature space and establish the comparison theorem.By using the method of operator estimation,the error bound and learning rate of the learning problem are obtained,namely,Theorem 1,Theorem 2 and Corollary 4.2.1.The content of this article is mainly divided into the following five parts:In chapter one,the development history of statistical learning theory is reviewed and its basic framework is constructed.Chapter two introduces some basic knowledge of the kernel method,the definitions and properties of which are used in the following.Chapter three first costructs the algorithm of kernel canonical correlation analysis,and then introduces the principle and algorithm of principal component analysis.Among them,the construction of the kernel canonical correlation analysis algorithm using the covariance operator sets the stage for the following construction of the learning algorithm of kernel-based principal component analysis.Chapter four gets the error bound and learning rate of the kernel-based principal component analysis learning algorithm.Firstly,the learning algorithm of kernel principal component analysis is proposed using the covariance operator.Finally,for the uncentered case,the operator estimation method is used to obtain the learning rate and error bound of this algorithmIn chapter five,the main results of this paper are summarized and the next research direction is given.