KERNEL METHODS FOR PRINCIPAL COMPONENT ANALYSIS (PCA) A comparative study of classical and kernel pca (original) (raw)
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Principal Component Analysis -A Tutorial
Dimensionality reduction is one of the preprocessing steps in many machine learning applications and it is used to transform the features into a lower dimension space. Principal Component Analysis (PCA) technique is one of the most famous unsupervised dimensionality reduction techniques. The goal of the PCA is to find the space, which represents the direction of the maximum variance of the given data. This paper highlights the basic background needed to understand and implement the PCA technique. This paper starts with basic definitions of the PCA technique and the algorithms of two methods of calculating PCA, namely, the covariance matrix and Singular Value Decomposition (SVD) methods. Moreover, a number of numerical examples are illustrated to show how the PCA space is calculated in easy steps. Three experiments are conducted to show how to apply PCA in the real applications including biometrics, image compression, and visualization of high-dimensional datasets.
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We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space transforms. Our results extend earlier work for probabilistic Karhunen-Loève transforms on compression of wavelet images. Our object is algorithms for optimization, selection of efficient bases, or components, which serve to minimize entropy and error; and hence to improve digital representation of images, and hence of optimal storage, and transmission. We prove several new theorems for data-dimension reduction. Moreover, with the use of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify when a choice of Gaussian kernel is optimal. Contents Data and digital image illustrations 1. Introduction 2. Karhunen-Loève transform or Principal Component Analysis 2.1. The Algorithm for a Digital Image or Data Application 2.2. Principal Component Analysis in a Digital Image 2.3. Dimension Reduction and Principal Component Analysis 2.4.
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Mercer kernels are used for a wide range of image and signal processing tasks like de-noising, clustering, discriminant analysis etc. These algorithms construct their solutions in terms of the expansions in a high-dimensional feature space F. However, many applications like kernel PCA (principal component analysis) can be used more effectively if a pre-image of the projection in the feature space is available. In this paper, we propose a novel method to reconstruct a unique approximate pre-image of a feature vector and apply it for statistical shape analysis. We provide some experimental results to demonstrate the advantages of kernel PCA over linear PCA for shape learning, which include, but are not limited to, ability to learn and distinguish multiple geometries of shapes and robustness to occlusions.
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In this letter, we present a simple and straightforward primal-dual support vector machine formulation to the problem of principal component analysis (PCA) in dual variables. By considering a mapping to a high-dimensional feature space and application of the kernel trick (Mercer theorem) kernel PCA is obtained as introduced by Schölkopf et al. While least squares support vector machine classifiers have a natural link with kernel Fisher discriminant analysis (minimizing the within class scatter around targets +1 and 1), for PCA analysis one can take the interpretation of a one-class modeling problem with zero target value around which one maximizes the variance. The score variables are interpreted as error variables within the problem formulation. In this way primal-dual constrained optimization problem interpretations to linear and kernel PCA analysis are obtained in a similar style as for least square-support vector machine (LS-SVM) classifiers. Index Terms-Kernel methods, kernel principal component analysis (PCA), least squares-support vector machine (LS-SVM), PCA analysis, SVMs.
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An approach to non-linear principal components using radially symmetric kernel basis functions is described. The procedure consists of two steps: a projection of the data set to a reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalized symmetric eigenvector equation. This is achieved by demanding a maximum variance transformation subject to a normalization
Journal of Science and Arts
Principal component analysis is commonly used as a pre-step before employing a classifier to avoid the negative effect of the dimensionality and multicollinearity. The performance of a classifier is severely affected by the deviations from the linearity of the data structure and noisy samples. In this paper, we propose a new classification system that overcomes the drawback of these crucial problems, simultaneously. Our proposal is relying on the kernel principal component analysis with a proper parameter selection approach with data complexity measures. According to the empirical results, F1, T2 and T3 in AUC, T3 in GMEAN and T2 and T3 in MCC performed better than classical and other complexity measures. Comparison of classifiers showed that Radial SVM performs better in AUC, and KNN performs better in GMEAN and MCC using KPCA with complexity measures. As a result, our proposed system produces better results in various classification algorithms with respect to classical approach.