User talk:L33th4x0r/math

$$ \alpha_1\boldsymbol{\phi}(\boldsymbol{x}_1) + \cdots + \alpha_n\boldsymbol{\phi}(\boldsymbol{x}_n) $$

$$ \approx \alpha_1\left[\frac{K(\boldsymbol{x}_1, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) + \cdots + \frac{K(\boldsymbol{x}_1, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d)\right] $$

$$ \alpha_l\left[\frac{K(\boldsymbol{x}_l, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) + \cdots + \frac{K(\boldsymbol{x}_l, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d)\right] $$

$$ = \alpha_1\frac{K(\boldsymbol{x}_1, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) + \cdots + \alpha_1\frac{K(\boldsymbol{x}_1, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d) $$

$$ \alpha_l\frac{K(\boldsymbol{x}_l, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) + \cdots + \alpha_l\frac{K(\boldsymbol{x}_l, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d) $$

$$ = \alpha_1\frac{K(\boldsymbol{x}_1, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) + \cdots + \alpha_l\frac{K(\boldsymbol{x}_l, \boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\boldsymbol{\phi}(\boldsymbol{z}_1) $$

$$ \alpha_1\frac{K(\boldsymbol{x}_1, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d) + \cdots + \alpha_l\frac{K(\boldsymbol{x}_l, \boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\boldsymbol{\phi}(\boldsymbol{z}_d)

$$

$$ = \left[\alpha_1\frac{K(\boldsymbol{x}_1,\boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2} + \cdots + \alpha_l\frac{K(\boldsymbol{x}_l,\boldsymbol{z}_1)}{|\boldsymbol{\phi}(\boldsymbol{z}_1)|^2}\right]\boldsymbol{\phi}(\boldsymbol{z}_1) $$

$$ \left[\alpha_1\frac{K(\boldsymbol{x}_1,\boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2} + \cdots + \alpha_l\frac{K(\boldsymbol{x}_l,\boldsymbol{z}_d)}{|\boldsymbol{\phi}(\boldsymbol{z}_d)|^2}\right]\boldsymbol{\phi}(\boldsymbol{z}_d) $$

$$ \boldsymbol{\phi}_c(\boldsymbol{x}) = \begin{cases} (1,0) & \mbox{if } \boldsymbol{x} \notin X' \\ (-1/2, \sqrt{3}/2) & \mbox{if } \boldsymbol{x} \in X' \mbox{ and } c(\boldsymbol{x}) = 1 \\ (-1/2, -\sqrt{3}/2) & \mbox{if } \boldsymbol{x} \in X' \mbox{ and } c(\boldsymbol{x}) = -1 \end{cases} $$

$$ K_c(\boldsymbol{x}, \boldsymbol{y}) = \begin{cases} 1 & \mbox{if } \boldsymbol{x}, \boldsymbol{y} \notin X' \mbox{ or } [\boldsymbol{x}, \boldsymbol{y} \in X' \mbox{ and } c(\boldsymbol{x}) = c(\boldsymbol{y})] \\ -1/2 & \mbox{otherwise} \end{cases} $$

$$ K(\boldsymbol{x}, \boldsymbol{z}) = \exp\left(-\frac{||\boldsymbol{x}-\boldsymbol{z}||^2}{2\sigma^2}\right) $$