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Kernel Methods

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Springer Handbook of Computational Intelligence

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Abstract

This chapter addresses the study of kernel methods, a class of techniques that play a major role in machine learning and nonparametric statistics. Among others, these methods include support vector machines (GlossaryTerm

SVM

s) and least squares GlossaryTerm

SVM

s, kernel principal component analysis, kernel Fisher discriminant analysis, and Gaussian processes. The use of kernel methods is systematic and properly motivated by statistical principles. In practical applications, kernel methods lead to flexible predictive models that often outperform competing approaches in terms of generalization performance. The core idea consists of mapping data into a high-dimensional space by means of a feature map. Since the feature map is normally chosen to be nonlinear, a linear model in the feature space corresponds to a nonlinear rule in the original domain. This fact suits many real world data analysis problems that often require nonlinear models to describe their structure.

In Sect. 32.1 we present historical notes and summarize the main ingredients of kernel methods. In Sect. 32.2 we present the core ideas of statistical learning and show how regularization can be employed to devise practical learning algorithms. In Sect. 32.3 we show a selection of techniques that are representative of a large class of kernel methods; these techniques – termed primal–dual methods – use Lagrange duality as the main mathematical tools. Section 32.4 discusses Gaussian processes, a class of kernel methods that uses a Bayesian approach to perform inference and learning. Section 32.5 recalls different approaches for the tuning of parameters. In Sect. 32.6 we review the mathematical properties of different yet equivalent notions of kernels and recall a number of specialized kernels for learning problems involving structured data. We conclude the chapter by presenting applications in Sect. 32.7.

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Abbreviations

ERM:

empirical risk minimization

GACV:

generalized approximate cross-validation

GP:

Gaussian process

HS:

Hilbert space

i.i.d.:

independent, identically distributed

KKT:

Karush–Kuhn–Tucker

LASSO:

least absolute shrinkage and selection operator

LOO:

leave-one-out

LS:

least square

MAP:

maximum a posteriori

MEG:

magnetoencephalography

MKL:

multiple kernel learning

ML:

maximum likelihood

MLP:

multilayer perceptron

PCA:

principal component analysis

QP:

quadratic programming

r.k.:

reproducing kernel

RBF:

radial basis function

RKHS:

reproducing kernel Hilbert space

SMO:

sequential minimum optimization

SRM:

structural risk minimization

SVC:

support vector classification

SVD:

singular value decomposition

SVM:

support vector machine

VC:

Vapnik–Chervonenkis

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  1. Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001, Leuven, Belgium

    Marco Signoretto & Johan A. K. Suykens

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  1. Systems Research Inst., Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

  2. Dep. Electrical and Computer Engineering, University of Alberta, 116 Street 9107, T6J 2V4, Edmonton, Alberta, Canada

    Witold Pedrycz

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Signoretto, M., Suykens, J.A.K. (2015). Kernel Methods. In: Kacprzyk, J., Pedrycz, W. (eds) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43505-2_32

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