this article with the regression case. To explain this, we will introduce a suitable definition of a margin that is maximized in both cases

In kernel methods, all the information about the training data is contained in the Gram matrix. If this matrix has large diagonal values, which arises for many types of kernels, then kernel methods do not perform well. We propose and test several methods for dealing with this problem by reducing the dynamic range of the matrix while preserving the positive definiteness of the Hessian of the quadratic programming problem that one has to solve when training a Support Vector Machine.

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Contents Preface vii 1 Introduction to Large Margin Classifiers 1 Alex J. Smola, Peter Bartlett, Bernhard Scholkopf, and Dale Schuurmans 2 Large Margin Rank Boundaries for Ordinal Regression 29 Ralf Herbrich, Thore Graepel, and Klaus Obermayer References 46 Smola, Bartlett, Scholkopf, and Schuurmans: Advances in Large Margin Classifiers 1999/07/12 09:57 Preface Some good quote who knows some clever stuff ... and some more visionary comments Alexander J. Smola, Peter Bartlett, Bernhard Scholkopf, Dale Schuurmans Berlin, Canberra, Waterloo, July 1999 Smola, Bartlett, Scholkopf, and Schuurmans: Advances in Large Margin Classifiers 1999/07/12 09:57 1 Introduction to Large Margin Classifiers The aim of this chapter is to provide a brief introduction to the basic concepts of large margin classifiers for readers unfamiliar with the topic. Moreover it is aimed at establishing a common ba

We show that, from the topological point of view, 2-tape Büchi automata have the same accepting power as Turing machines equipped with a Büchi acceptance condition. The Borel and the Wadge hierarchies of the class RATω of infinitary rational relations accepted by 2-tape Büchi automata are equal to the Borel and the Wadge hierarchies of ω-languages accepted by realtime Büchi 1-counter automata or by Büchi Turing machines. In particular, for every non null recursive ordinal α, there exist some Σ 0 α-complete and some Π 0 α-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal γ 1 2 which is strictly greater than the first non recursive ordinal ω CK 1. This very surprising result gives answers to questions of Simonnet [Sim92] and of Lescow and Thomas [Tho88, LT94].

The two-spiral task is a well-known benchmark for binary classification. The data consist of points on two intertwined spirals which cannot be linearly separated. This article reviews how this task and some of its variations have significantly inspired the development of several important methods in the history of artificial neural networks. The two-spiral task became popular for several different reasons: 1) It was regarded as extremely challenging; 2) It belonged to a suite of standard benchmark tasks; 3) It had visual appeal and was convenient to use in pilot studies. The article also presents an example which demonstrates how small variations of the two-spiral task such as relative rotations of the two spirals can lead to qualitatively different generalisation results.

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