# 径向基函数核

${\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {||\mathbf {x} -\mathbf {x'} ||_{2}^{2}}{2\sigma ^{2}}}\right)}$

${\displaystyle \textstyle ||\mathbf {x} -\mathbf {x'} ||_{2}^{2}}$可以看做两个特征向量之间的平方欧几里得距离${\displaystyle \sigma }$是一个自由参数。一种等价但更为简单的定义是设一个新的参数${\displaystyle \gamma }$，其表达式为${\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}}$

${\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma ||\mathbf {x} -\mathbf {x'} ||_{2}^{2})}$

${\displaystyle \exp \left(-{\frac {1}{2}}||\mathbf {x} -\mathbf {x'} ||_{2}^{2}\right)=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}||\mathbf {x} ||_{2}^{2}\right)\exp \left(-{\frac {1}{2}}||\mathbf {x'} ||_{2}^{2}\right)}$

## 近似

${\displaystyle z(\mathbf {x} )z(\mathbf {x'} )\approx \varphi (\mathbf {x} )\varphi (\mathbf {x'} )=K(\mathbf {x} ,\mathbf {x'} )}$

## 参考资料

1. ^ Yin-Wen Chang, Cho-Jui Hsieh, Kai-Wei Chang, Michael Ringgaard and Chih-Jen Lin (2010). Training and testing low-degree polynomial data mappings via linear SVM. J. Machine Learning Research 11: 1471–1490.
2. Vert, Jean-Philippe, Koji Tsuda, and Bernhard Schölkopf (2004). "A primer on kernel methods." Kernel Methods in Computational Biology.
3. ^ Shashua, Amnon. Introduction to Machine Learning: Class Notes 67577. 2009. arXiv:0904.3664 [cs.LG].
4. ^ Andreas Müller (2012). Kernel Approximations for Efficient SVMs (and other feature extraction methods).
5. ^ Ali Rahimi and Benjamin Recht (2007). Random features for large-scale kernel machines. Neural Information Processing Systems.