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- Bernhard Schölkopf, Alexander J. Smola, Klaus-Robert Müller
- Neural Computation
- 1998

A new method for performing a nonlinear form of principal component analysis is proposed. By the use of integral operator kernel functions, one can efficiently compute principal components in high-dimensional feature spaces, related to input space by some nonlinear map—for instance, the space of all possible five-pixel products in 16 × 16 images. We give… (More)

- Bernhard Schölkopf, John C. Platt, John Shawe-Taylor, Alexander J. Smola, Robert C. Williamson
- Neural Computation
- 2001

Suppose you are given some data set drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified value between 0 and 1. We propose a method to approach this problem by trying to estimate a function f… (More)

- Alexander J. Smola, Bernhard Schölkopf
- Statistics and Computing
- 2004

In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications… (More)

A new method for performing a nonlinear form of Principal Component Analysis is proposed. By the use of integral operator kernel functions, one can eeciently compute principal components in highh dimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible dpixel products in images. We give the derivation… (More)

- Bernhard Schölkopf, Alexander J. Smola, Robert C. Williamson, Peter L. Bartlett
- Neural Computation
- 2000

We describe a new class of Support Vector algorithms for regression and classiication. In these algorithms, a parameter lets one eectively control the number of Support Vectors. While this can be useful in its own right, the parametrization has the additional beneet of enabling us to eliminate one of the other free parameters of the algorithm: the accuracy… (More)

We propose a framework for analyzing and comparing distributions, allowing us to design statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS). We present two tests based on large deviation… (More)

A new regression technique based on Vapnik's concept of support vectors is introduced. We compare support vector regression (SVR) with a committee regression technique (bagging) based on regression trees and ridge regression done in feature space. On the basis of these experiments, it is expected that SVR will have advantages in high dimensionality space… (More)

We propose an independence criterion based on the eigen-spectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous… (More)

- Bernhard Schölkopf, Sebastian Mika, +4 authors Alexander J. Smola
- IEEE Trans. Neural Networks
- 1999

This paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector (SV) kernel functions. We first discuss the geometry of feature space. In particular, we review what is known about the shape of the image of input space under the feature space map, and how this influences the capacity of SV methods.… (More)