The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.

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

This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classication tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance vector machine' (RVM), a model of identical functional form to the popular and state-of-the-art `support vector machine' (SVM). We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while oering a number of additional advantages. These include the benets of probabilistic predictions, automatic estimation of `nuisance' parameters, and the facility to utilise arbitrary basis functions (e.g. non-`Mercer' kernels).

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 publication can be retrieved by anonymous ftp to publications.ai.mit.edu. The pathname for this publication is: ai-publications/1500-1999/AIM-1606.ps.Z This paper shows a relationship between two di erent approximation techniques: the Support Vector Machines (SVM), proposed by V.Vapnik (1995), and a sparse approximation scheme that resembles the Basis Pursuit De-Noising algorithm (Chen, 1995 � Chen, Donoho and Saunders, 1995). SVM is a technique which can be derived from the Structural Risk Minimization Principle (Vapnik, 1982) and can be used to estimate the parameters of several di erent approximation schemes, including Radial Basis Functions, algebraic/trigonometric polynomials, B-splines, and some forms of Multilayer Perceptrons. Basis Pursuit De-Noising is a sparse approximation technique, in which a function is reconstructed by using a small number of basis functions chosen from a large set (the dictionary). We show that, if the data are noiseless, the modi ed version of Basis Pursuit De-Noising proposed in this paper is equivalent to SVM in the following sense: if applied to the same data set the two techniques give the same solution, which is obtained by solving the same quadratic programming problem. In the appendix we also present a derivation of the SVM technique in the framework of regularization theory, rather than statistical learning theory, establishing a connection between SVM, sparse approximation and regularization theory.

We present a new method that we call Generalized Discriminant Analysis (GDA) to deal with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the Support Vector Machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high dimensional feature space. In the transformed space, linear properties make it easy to extend and generalize the classical Linear Discriminant Analysis (LDA) to non linear discriminant analysis. The formulation is expressed as an eigenvalue problem resolution. Using a different kernel, one can cover a wide class of nonlinearities. For both simulated data and alternate kernels, we give classification results as well as the shape of the separating function. The results are confirmed using a real data to perform seed classification. 1. Introduction Linear discriminant analysis (LDA) is a traditional statistical method which has proven successful on classification problems [Fukunaga, 1990]. The p...

In kernel based methods such as Regularization Networks large datasets pose signi- cant problems since the number of basis functions required for an optimal solution equals the number of samples. We present a sparse greedy approximation technique to construct a compressed representation of the design matrix. Experimental results are given and connections to Kernel-PCA, Sparse Kernel Feature Analysis, and Matching Pursuit are pointed out. 1. Introduction Many recent advances in machine learning such as Support Vector Machines [Vapnik, 1995], Regularization Networks [Girosi et al., 1995], or Gaussian Processes [Williams, 1998] are based on kernel methods. Given an m-sample f(x 1 ; y 1 ); : : : ; (x m ; y m )g of patterns x i 2 X and target values y i 2 Y these algorithms minimize the regularized risk functional min f2H R reg [f ] = 1 m m X i=1 c(x i ; y i ; f(x i )) + 2 kfk 2 H : (1) Here H denotes a reproducing kernel Hilbert space (RKHS) [Aronszajn, 1950],...

The problem of empirical data modelling is germane to many engineering applications. In empirical data modelling a process of induction is used to build up a model of the system, from which it is hoped to deduce responses of the system that have yet to be observed. Ultimately the quantity and quality of the observations govern the performance of this empirical model. By its observational nature data obtained is finite and sampled; typically this sampling is non-uniform and due to the high dimensional nature of the problem the data will form only a sparse distribution in the input space. Consequently the problem is nearly always ill posed (Poggio et al., 1985) in the sense of Hadamard (Hadamard, 1923). Traditional neural network approaches have suffered difficulties with generalisation, producing models that can overfit the data. This is a consequence of the optimisation algorithms used for parameter selection and the statistical measures used to select the ’best’ model. The foundations of Support Vector Machines (SVM) have been developed by Vapnik (1995) and are gaining popularity due to many attractive features, and promising empirical performance. The formulation embodies the Structural Risk Minimisation (SRM) principle, which has been shown to be superior, (Gunn et al., 1997), to traditional Empirical Risk Minimisation (ERM) principle, employed by conventional neural networks. SRM minimises an upper bound on the expected risk, as opposed to ERM that minimises the error on the training data. It is this difference which equips SVM with a greater ability to generalise, which is the goal in statistical learning. SVMs were developed to solve the classification problem, but recently they have been extended to the domain of regression problems (Vapnik et al., 1997). In the literature the terminology for SVMs can be slightly confusing. The term SVM is typically used to describe classification with support vector methods and support vector regression is used to describe regression with support vector methods. In this report the term SVM will refer to both classification and regression methods, and the terms Support Vector Classification (SVC) and Support Vector Regression (SVR) will be used for specification. This section continues with a brief introduction to the structural risk

Support Vector Learning Machines (SVM) are finding application in pattern recognition, regression estimation, and operator inversion for ill-posed problems. Against this very general backdrop, any methods for improving the generalization performance, or for improving the speed in test phase, of SVMs are of increasing interest. In this paper we combine two such techniques on a pattern recognition problem. The method for improving generalization performance (the "virtual support vector" method) does so by incorporating known invariances of the problem. This method achieves a drop in the error rate on 10,000 NIST test digit images of 1.4% to 1.0%. The method for improving the speed (the "reduced set" method) does so by approximating the support vector decision surface. We apply this method to achieve a factor of fifty speedup in test phase over the virtual support vector machine. The combined approach yields a machine which is both 22 times faster than the original machine, and which has ...

In this paper a correspondence is derived between regularization operators used in Regularization Networks and Support Vector Kernels. We prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. Moreover the paper provides an analysis of currently used Support Vector Kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of Radial Basis Functions, namely conditionally positive definite functions, may be used as Support Vector kernels.