This procedure consists on substituting the inner product in the space of input variables by an appropriate function k(x,x’) that associates to each two inputs an real number real that is k(x,x’) ? R, for any x,x’ ? X. Thus, this is a method for converting a linear classifier algorithm into a non-linear one by using a non-linear function to map the original observations into a higher-dimensional space; this makes a linear classification in the new space equivalent to non-linear classification in the original space.
Published in Chapter:
Support Vector Machines
Cecilio Angulo (Technical University of Catalonia, Spain) and Luis Gonzalez-Abril (Technical University of Catalonia, Spain)
Copyright: © 2009
|Pages: 6
DOI: 10.4018/978-1-59904-849-9.ch222
Abstract
Support Vector Machines -- SVMs -- are learning machines, originally designed for bi-classification problems, implementing the well-known Structural Risk Minimization (SRM) inductive principle to obtain good generalization on a limited number of learning patterns (Vapnik, 1998). The optimization criterion for these machines is maximizing the margin between two classes, i.e. the distance between two parallel hyperplanes that split the vectors of each one of the two classes, since larger is the margin separating classes, smaller is the VC dimension of the learning machine, which theoretically ensures a good generalization performance (Vapnik, 1998), as it has been demonstrated in a number of real applications (Cristianini, 2000). In its formulation is applicable the kernel trick, which improves the capacity of these algorithms, learning not being directly performed in the original space of data but in a new space called feature space; for this reason this algorithm is one of the most representative of the called Kernel Machines (KMs). Main theory was originally developed on the sixties and seventies by V. Vapnik and A. Chervonenkis (Vapnik et al., 1963, Vapnik et al., 1971, Vapnik, 1995, Vapnik, 1998), on the basis of a separable binary classification problem, however generalization in the use of these learning algorithms did not take place until the nineties (Boser et al., 1992). SVMs has been used thoroughly in any kind of learning problems, mainly in classification problems, although also in other problems like regression (Schölkopf et al., 2004) or clustering (Ben-Hur et al., 2001). The fields of Optic Character Recognition (Cortes et al., 1995) and Text Categorization (Sebastiani, 2002) were the most important initial applications where SVMs were used. With the extended application of new kernels, novel applications have taken place in the field of Bioinformatics, concretely many works are related with the classification of data in Genetic Expression (Microarray Gene Expression) (Brown et al., 1997) and detecting structures between proteins and their relationship with the chains of DNA (Jaakkola et al., 2000). Other applications include image identification, voice recognition, prediction in time series, etc. A more extensive list of applications can be found in (Guyon, 2006).