The paper introduces a new method for performing nonlinear Principal Component Analysis (PCA) using kernel functions. The authors, Bernhard Schölkopf, Alexander Smola, and Klaus-Robert Müller, propose a technique that allows efficient computation of principal components in high-dimensional feature spaces, even when the feature space is nonlinearly mapped from the input space. The method leverages integral operator kernel functions, similar to those used in Support Vector Machines (SVMs), to handle the nonlinear transformation without explicitly computing the mapping. The paper provides a derivation of the method and presents experimental results on polynomial feature extraction for pattern recognition. The key steps include mapping the data to a feature space, computing the covariance matrix in this space, and solving the eigenvalue problem using kernel functions to avoid explicit computation of the nonlinear transformation. The authors also discuss various types of kernels, such as polynomial, radial basis functions, and sigmoid kernels, and their applications in different scenarios.The paper introduces a new method for performing nonlinear Principal Component Analysis (PCA) using kernel functions. The authors, Bernhard Schölkopf, Alexander Smola, and Klaus-Robert Müller, propose a technique that allows efficient computation of principal components in high-dimensional feature spaces, even when the feature space is nonlinearly mapped from the input space. The method leverages integral operator kernel functions, similar to those used in Support Vector Machines (SVMs), to handle the nonlinear transformation without explicitly computing the mapping. The paper provides a derivation of the method and presents experimental results on polynomial feature extraction for pattern recognition. The key steps include mapping the data to a feature space, computing the covariance matrix in this space, and solving the eigenvalue problem using kernel functions to avoid explicit computation of the nonlinear transformation. The authors also discuss various types of kernels, such as polynomial, radial basis functions, and sigmoid kernels, and their applications in different scenarios.