In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.[1]
The RBF kernel on two samples and , represented as feature vectors in some input space, is defined as[2]
may be recognized as the squared Euclidean distance between the two feature vectors. is a free parameter. An equivalent definition involves a parameter :
Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure.[2]
The feature space of the kernel has an infinite number of dimensions; for , its expansion using the multinomial theorem is:[3]
where ,
^Chang, Yin-Wen; Hsieh, Cho-Jui; Chang, Kai-Wei; Ringgaard, Michael; Lin, Chih-Jen (2010). "Training and testing low-degree polynomial data mappings via linear SVM". Journal of Machine Learning Research. 11: 1471–1490.
^ abJean-Philippe Vert, Koji Tsuda, and Bernhard Schölkopf (2004). "A primer on kernel methods". Kernel Methods in Computational Biology.
^Shashua, Amnon (2009). "Introduction to Machine Learning: Class Notes 67577". arXiv:0904.3664v1 [cs.LG].
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