Radial basis function kernel: Difference between revisions

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In [[machine learning]], the '''[[radial basis function]] kernel''', or '''RBF kernel''', is a popular [[Positive-definite kernel|kernel function]] used in various [[kernel method|kernelized]] learning algorithms. In particular, it is commonly used in [[support vector machine]] [[statistical classification|classification]].<ref name="Chang2010">{{cite journal | last1 = Chang | first1 = Yin-Wen | last2 = Hsieh | first2 = Cho-Jui | last3 = Chang | first3 = Kai-Wei | last4 = Ringgaard | first4 = Michael | last5 = Lin | first5 = Chih-Jen | year = 2010 | title = Training and testing low-degree polynomial data mappings via linear SVM | url = http://jmlr.org/papers/v11/chang10a.html | journal = Journal of Machine Learning Research | volume = 11 | pages = 1471–1490 }}</ref>

The RBF kernel on two samples <math>\mathbf{x}\in \mathbb{R}^{k}</math> and '''<math>\mathbf{x''''}</math>, represented as feature vectors in some ''input space'', is defined as<ref name="primer">Jean-Philippe Vert, Koji Tsuda, and Bernhard Schölkopf (2004). [http://cbio.ensmp.fr/~jvert/publi/04kmcbbook/kernelprimer.pdf "A primer on kernel methods".] ''Kernel Methods in Computational Biology''.</ref>

:<math>K(\mathbf{x}, \mathbf{x'}) = \exp\left(-\frac{\|\mathbf{x} - \mathbf{x'}\|^2}{2\sigma^2}\right)</math>