Kernel based algorithms such as support vector machines have achieved considerable success in various problems in the batch setting where all of the training data is available in advance. Support vector machines combine the so-called kernel trick with the large margin idea. There has been little use of these methods in an online setting suitable for real-time applications. In this paper we consider online learning in a Reproducing Kernel Hilbert Space. By considering classical stochastic gradient descent within a feature space, and the use of some straightforward tricks, we develop simple and computationally efficient algorithms for a wide range of problems such as classification, regression, and novelty detection. In addition to allowing the exploitation of the kernel trick in an online setting, we examine the value of large margins for classification in the online setting with a drifting target. We derive worst case loss bounds and moreover we show the convergence of the hypothesis to the minimiser of the regularised risk functional. We present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection. In addition

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 and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.

This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and

Editor: Kernel based methods are increasingly being used for data modeling because of their conceptual simplicity and outstanding performance on many tasks. However, the kernel function is often chosen using trial-and-error heuristics. In this paper we address the problem of measuring the degree of agreement between a kernel and a learning task. A quantitative measure of agreement is important from both a theoretical and practical point of view. We propose a quantity to capture this notion, which we call Alignment. We study its theoretical properties, and derive a series of simple algorithms for adapting a kernel to the labels and vice versa. This produces a series of novel methods for clustering and transduction, kernel combination and kernel selection. The algorithms are tested on two publicly available datasets and are shown to exhibit good performance.

Kernel PCA as a nonlinear feature extractor has proven powerful as a preprocessing step for classification algorithms. But it can also be considered as a natural generalization of linear principal component analysis. This gives rise to the question how to use nonlinear features for data compression, reconstruction, and de-noising, applications common in linear PCA. This is a nontrivial task, as the results provided by kernel PCA live in some high dimensional feature space and need not have pre-images in input space. This work presents ideas for finding approximate pre-images, focusing on Gaussian kernels, and shows experimental results using these pre-images in data reconstruction and de-noising on toy examples as well as on real world data.

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. Following this, we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel, using the example of the class of inhomogeneous polynomial kernels, which are often used in SV pattern recognition. We then discuss the connection between feature space and input space by dealing with the question of how one can, given some vector in feature space, find a preimage (exact or approximate) in input space. We describe algorithms to tackle this issue, and show their utility in two applications of kernel methods. First, we use it to reduce the computational complexity of SV decision functions; second, we combine it with the Kernel PCA algorithm, thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real-world data.

We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinite-dimensional unit ball in feature space into a finite-dimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence, we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines.

Kernels are problem-specific functions that act as an interface between the learning system and the data. While it is well-known when the combination of two kernels is again a valid kernel, it is an open question if the resulting kernel will perform well. In particular, in which situations can a combination of kernel be expected to perform better than its components considered separately? We investigate this problem by looking at the task of designing kernels for hypertext classification, where both words and links information can be exploited. We provide sufficient conditions that indicate when an improvement can be expected, highlighting and formalising the notion of "independent kernels". Experimental results confirm the predictions of the theory in the hypertext domain.

Algorithms based on Mercer kernels construct their solutions in terms of expansions in a high-dimensional feature space F . Previous work has shown that all algorithms which can be formulated in terms of dot products in F can be performed using a kernel without explicitly working in F . The list of such algorithms includes support vector machines and nonlinear kernel principal component extraction. So far, however, it did not include the reconstruction of patterns from their largest nonlinear principal components, a technique which is common practice in linear principal component analysis. The present work proposes an idea for approximately performing this task. As an illustrative example, an application to the de-noising of data clusters is presented. 1 Kernels and Feature Spaces A Mercer kernel is a function k(x; y) which for all data sets fx 1 ; : : : ; x ` g ae R N gives rise to a positive (not necessarily definite) matrix K ij := k(x i ; x j ) [4]. One can show that using k ins...

We incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination. Employing a unified framework in terms of a nonlinear variant of the Rayleigh coefficient, we propose non-linear generalizations of Fisher's discriminant and oriented PCA using Support Vector kernel functions.