Radial basis function kernel: Difference between revisions

Revision as of 08:55, 30 September 2014

Revision as of 08:45, 7 November 2014

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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".] ''Kernel Methods in Computational Biology''.</ref>

:<math>K(\mathbf{x}, \mathbf{x'}) = \exp\left(-\frac{||\mathbf{x} - \mathbf{x'}||_2^2}{2\sigma^2}\right)</math>

<math>\textstyle||\mathbf{x} - \mathbf{x'}||_2^2</math> may be recognized as the [[Euclidean distance#Squared Euclidean distance|squared Euclidean distance]] between the two feature vectors. <math>\sigma</math> is a free parameter. An equivalent, but simpler, definition involves a parameter <math>\textstyle\gamma = -\tfrac{1}{2\sigma^2}</math>:

:<math>K(\mathbf{x}, \mathbf{x'}) = \exp(\gamma||\mathbf{x} - \mathbf{x'}||_2^2)</math>

Since the value of the RBF kernel decreases with distance and ranges between zero (in the limit) and one (when {{math|'''x''' {{=}} '''x''''}}), it has a ready interpretation as a [[similarity measure]].<ref name="primer"/>

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}}</ref>

:<math>\exp\left(-\frac{1}{2}||\mathbf{x} - \mathbf{x'}||_2^2\right) = \sum_{j=0}^\infty \frac{(\mathbf{x}^\top \mathbf{x'})^j}{j!} \exp\left(-\frac{1}{2}||\mathbf{x}||_2^2\right)

\exp\left(-\frac{1}{2}||\mathbf{x'}||_2^2\right)</math>

==Approximations==