The Nystr "om methods have been popular techniques for scalable kernel based learning. They approximate explicit, low-dimensional feature mappings for kernel functions from the pairwise comparisons with the training data. However, Nystr "om methods are generally applied without the supervision provided by the training labels in the classification regression problems. This leads to pairwise comparisons with randomly chosen training samples in the model. Conversely, this work studies a supervised Nystr "om method that chooses the critical subsets of samples for the success of the Machine Learning model. Particularly, we select the Nystr "om support vectors via the negative margin criterion, and create explicit feature maps that are more suitable for the classification task on the data. Experimental results on six datasets show that, without increasing the complexity over unsupervised techniques, our method can significantly improve the classification performance achieved via kernel approximation methods and reduce the number of features needed to reach or exceed the performance of the full-dimensional kernel machines.
 Kernel principal component analysis has been introduced as a method of extracting a set of orthonormal nonlinear features from multivariate data, and many impressive applications are being reported within the literature. This article presents the view that the eigenvalue decomposition of a kernel matrix can also provide the discrete expansion coefficients required for a nonparametric orthogonal series density estimator. In addition to providing novel insights into nonparametric density estimation, this article provides an intuitively appealing interpretation for the nonlinear features extracted from data using kernel principal component analysis. A crucial technique for scaling kernel methods to very large data sets reaching or exceeding millions of instances is based on low-rank approximation of kernel matrices. We introduce a new family of algorithms based on mixtures of Nystrom approximations, ensemble Nystrom algorithms, that yield more accurate low-rank approximations than the standard Nystrom method. We give a detailed study of variants of these algorithms based on simple averaging, an exponential weight method, or regression-based methods. We also present a theoretical analysis of these algorithms, including novel error bounds guaranteeing a better convergence rate than the standard Nystrom method. Finally, we report results of extensive experiments with several data sets containing up to 1M points demonstrating the significant improvement over the standard Nystrom approximation.