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.

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...

The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems. PREDICTION WITH GAUSSIAN PROCESSES: FROM LINEAR REGRESSION TO LINEAR PREDICTION AND BEYOND 2 1 Introduction In the last decade neural networks have been used to tackle regression and classification problems, with some notable successes. It has also been widely recognized that they form a part of a wide variety of non-linear statistical techniques that can be used for...

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this paper we very briefly review some of these results. RKHS can be chosen tailored to the problem at hand in many ways, and we review a few of them, including radial basis function and smoothing spline ANOVA spaces. Girosi (1997), Smola and Scholkopf (1997), Scholkopf et al (1997) and others have noted the relationship between SVM's and penalty methods as used in the statistical theory of nonparametric regression. In Section 1.2 we elaborate on this, and show how replacing the likelihood functional of the logit (log odds ratio) in penalized likelihood methods for Bernoulli [yes-no] data, with certain other functionals of the logit (to be called SVM functionals) results in several of the SVM's that are of modern research interest. The SVM functionals we consider more closely resemble a "goodness-of-fit" measured by classification error than a "goodness-of-fit" measured by the comparative Kullback-Liebler distance, which is frequently associated with likelihood functionals. This observation is not new or profound, but it is hoped that the discussion here will help to bridge the conceptual gap between classical nonparametric regression via penalized likelihood methods, and SVM's in RKHS. Furthermore, since SVM's can be expected to provide more compact representations of the desired classification boundaries than boundaries based on estimating the logit by penalized likelihood methods, they have potential as a prescreening or model selection tool in sifting through many variables or regions of attribute space to find influential quantities, even when the ultimate goal is not classification, but to understand how the logit varies as the important variables change throughout their range. This is potentially applicable to the variable/model selection problem in demographic m...

Practical experience has shown that in order to obtain the best possible performance, prior knowledge about invariances of a classification problem at hand ought to be incorporated into the training procedure. We describe and review all known methods for doing so in support vector machines, provide experimental results, and discuss their respective merits. One of the significant new results reported in this work is our recent achievement of the lowest reported test error on the well-known MNIST digit recognition benchmark task, with SVM training times that are also significantly faster than previous SVM methods.

We consider the problem of assigning an input vector x to one of m classes by predicting P (cjx) for c = 1; : : : ; m. For a two-class problem, the probability of class 1 given x is estimated by oe(y(x)), where oe(y) = 1=(1 + e ). A Gaussian process prior is placed on y(x), and is combined with the training data to obtain predictions for new x points.

Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a self-contained proof utilizing the feature space associated with a kernel. The result shows that a wide range of problems have optimal solutions that live in the finite dimensional span of the training examples mapped into feature space, thus enabling us to carry out kernel algorithms independent of the (potentially infinite) dimensionality of the feature space.

A new region-based approach to nonrigid motion tracking is described. Shape is defined in terms of a deformable triangular mesh that captures object shape plus a color texture map that captures object appearance. Photometric variations are also modeled. Nonrigid shape registration and motion tracking are achieved by posing the problem as an energy-based, robust minimization procedure. The approach provides robustness to occlusions, wrinkles, shadows, and specular highlights. The formulation is tailored to take advantage of texture mapping hardware available in many workstations, PC's, and game consoles. This enables nonrigid tracking at speeds approaching video rate. 1 Introduction A key open problem in tracking is that of encoding and comparing shapes as they undergo nonrigid deformation. Simply providing robustness to nonrigid deformation is insufficient, because deformation often provides important information about how shapes are related. To make things worse, tracking must also ...