Effective methods of capacity control via uniform convergence bounds for function expansions have been largely limited to Support Vector machines, wheregood bounds are obtainable by the entropy number approach.

In this article we study the effects of introducing structure in the input distribution of the data to be learnt by a simple perceptron. We determine the learning curves within the framework of Statistical Mechanics.Stepwise generalization occurs as a function of the number of examples when the distribution of patterns is highly anisotropic. Although extremely simple, the model seems to capture therelevant features of a class of Support Vector Machines which was recently shown to present this behavior.

New functionals for parameter (model) selection of Support Vector Machines areintroduced based on the concepts of the span of support vectors and rescaling of the feature space. It is shown that using these functionals, onecan both predict the best choice of parameters of the model and the relative quality of performance for any value of parameter.

We give necessary and sufficient conditions for uniqueness of the support vector solution for the problems of pattern recognition and regression estimation, for a general class of cost functions. We show that if the solution is not unique, all support vectors are necessarily at bound, and we give some simple examples of non-unique solutions. Wenote that uniqueness of the primal (dual) solution does not necessarily imply uniqueness of the dual (primal) solution. We show how to compute the threshold b when the solution is unique, but when all support vectors are at bound, in which case the usual method for determining b does not work. 1 Introduction Support vector machines (SVMs) have attracted wide interest as a means to implement structuralrisk minimization for the problems of classification and regression estimation. The fact that training an SVM amounts to solving a convex quadratic programming problem means that the solution found is global, and that if it is not unique, then the set of global solutions is itself convex; furthermore, if the objective functionis strictly convex, the solution is guaranteed to be unique [1]1.

In fact, for many of the kernels used (not the polynomial kernels) like Gaussian rbf-kernels it can be shown [6] that SV machines are universal approximators. While this is advantageous in general, parametric models are useful techniques in their own right. Especially if one happens to have additional knowledge about the problem, it would be unwise not to take advantage of it. For instance it might be the case that the major properties of the data are described by a combination of a small set of linear independent basis functions {¢Jt (.),..., ¢n (.)}. Or one may want to correct the data for some (e.g.

Face recognition is different from classical pattern-recognition problems such as character recognition.

We will also introduce a regularization strategy (analogous to weight decay) into boosting. This strategy uses slack variables to achieve a soft margin (section 4). Numerical experiments show the validity of our regularization approach in section 5 and finally a brief conclusion is given. 2 AdaBoost Algorithm Let {ht(x): t 1,...,T} be an ensemble of T hypotheses defined on input vector x and e

SVMs have empirically been shown to give good generalization performance on a wide variety of problems. However, the use of SVMs is stilI limited to a small group of researchers. One possible reason is that training algorithms for SVMs are slow, especially for large problems. Another explanation is that SVM training algorithms are complex, subtle, and sometimes difficult to implement. This paper describes a new SVM learning algorithm that is easy to implement, often faster, and has better scaling properties than the standard SVM training algorithm. The new SVM learning algorithm is called Sequential Minimal Optimization (or SMO).

On the other hand, discriminative methods such as support vector machines enable us to construct flexible decision boundaries and often result in classification performance superior to that of the model based approaches. An ideal classifier should combine these two complementary approaches. In this paper, we develop a natural way of achieving this combination by deriving kernel functions for use in discriminative methods such as support vector machines from generative probability models.

We introduce a semi-supervised support vector machine (S3yM) method. Given a training set of labeled data and a working set of unlabeled data, S3YM constructs a support vector machine using both the training and working sets. We use S3 YM to solve the transduction problem using overall risk minimization (ORM) posed by Yapnik. The transduction problem is to estimate the value of a classification function at the given points in the working set. This contrasts with the standard inductive learning problem of estimating the classification function at all possible values and then using the fixed function to deduce the classes of the working set data.