In supervised learning, the learning machine is given a training set of examples (or inputs) with associated labels (or output values). Usually the examples are in the form of attribute vectors, so that the input space is a subset of ℝn. Once the attribute vectors are available, a number of sets of hypotheses could be chosen for the problem. Among these, linear functions are the best understood and simplest to apply. Traditional statistics and the classical neural networks literature have developed many methods for discriminating between two classes of instances using linear functions, as well as methods for interpolation using linear functions. These techniques, which include both efficient iterative procedures and theoretical analysis of their generalisation properties, provide the framework within which the construction of more complex systems will be developed in the coming chapters. In this chapter we review results from the literature that will be relevant to the study of Support Vector Machines. We will first discuss algorithms and issues of classification, and then we will move on to the problem of regression. Throughout this book, we will refer to learning machines using hypotheses that form linear combinations of the input variables as linear learning machines.

Importantly, we will show that in most cases such machines can be represented in a particularly useful form, which we will call the dual representation. This fact will prove crucial in later chapters. The important notions of margin and margin distribution are also introduced in this chapter. The classification results are all introduced for the binary or two-class case, and at the end of the chapter it is shown how to generalise them to multiple classes.