# 徑向基函數核

$K(\mathbf{x}, \mathbf{x'}) = \exp\left(-\frac{||\mathbf{x} - \mathbf{x'}||_2^2}{2\sigma^2}\right)$

$\textstyle||\mathbf{x} - \mathbf{x'}||_2^2$可以看做兩個特徵向量之間的平方歐幾里得距離$\sigma$是一個自由參數。一種等價但更為簡單的定義是設一個新的參數$\gamma$，其表達式為$\textstyle\gamma = -\tfrac{1}{2\sigma^2}$

$K(\mathbf{x}, \mathbf{x'}) = \exp(\gamma||\mathbf{x} - \mathbf{x'}||_2^2)$

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

## 近似

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

## 參考資料

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. ^ 2.0 2.1 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. arXiv:0904.36641 [cs.LG]. 2009.
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.