Radial basis function kernel: Difference between revisions

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In [[machine learning]], the ('''Gaussian''') '''[[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">Yin-Wen Chang, Cho-Jui Hsieh, Kai-Wei Chang, Michael Ringgaard and Chih-Jen Lin (2010). [http://jmlr.org/papers/v11/chang10a.html "Training and testing low-degree polynomial data mappings via linear SVM"]. ''J. Machine Learning Research'' '''11''':1471–1490.</ref>

The RBF kernel on two samples '''x''' and '''x'''', represented as feature vectors in some ''input space'', is defined as<ref name="primer">Vert, 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>

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where <math>\textstyle\varphi</math> is the implicit mapping embedded in the RBF kernel.

One way to construct such a ''z'' is to randomly sample from the [[Fourier transformation]] of the kernel.<ref>Ali Rahimi and Benjamin Recht (2007). [http://www.eecs.berkeley.edu/~brecht/papers/07.rah.rec.nips.pdf "Random features for large-scale kernel machines"]. ''Neural Information Processing Systems''.</ref> Another approach uses the [[Nyström method]] to approximate the [[eigendecomposition]] of the [[Gramian matrix|Gram matrix]] ''K'', using only a random sample of the training set.<ref>{{cite journal |authors=Williams, C.K.I. Williams and Seeger, M. Seeger|title=Using the Nyström method to speed up kernel machines |journal=Advances in Neural Information Processing Systems |year=2001 |url= http://papers.nips.cc/paper/1866-using-the-nystrom-method-to-speed-up-kernel-machines}}</ref>

==External links==