Content deleted Content added
mNo edit summary |
m edit citations - add urls |
||
Line 1:
In [[machine learning]], the ('''Gaussian''') '''[[radial basis function]] kernel''', or '''RBF kernel''', is a popular [[Positive-definite kernel|kernel function]] 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).
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, Koji Tsuda, and Bernhard Schölkopf (2004). [http://cbio.ensmp.fr/~jvert/publi/04kmcbbook/kernelprimer.pdf "A primer on kernel methods
:<math>K(\mathbf{x}, \mathbf{x'}) = \exp\left(-\frac{||\mathbf{x} - \mathbf{x'}||_2^2}{2\sigma^2}\right)</math>
Line 32:
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. and Seeger, M. |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==
| |||