Χρήστης:Iiirxs/Support vector machine

$({\vec {x}}_{1},y_{1}),\ldots ,({\vec {x}}_{n},y_{n}),$

${\vec {w}}\cdot {\vec {x}}-b=0,$

${\vec {w}}\cdot {\vec {x}}_{i}-b\geq 1$, αν $y_{i}=1$ ,

${\vec {w}}\cdot {\vec {x}}_{i}-b\leq -1$, αν $y_{i}=-1$ .

$y_{i}({\vec {w}}\cdot {\vec {x}}_{i}-b)\geq 1,\quad {\text{ for all }}1\leq i\leq n.\qquad \qquad (1)$

$\max \left(0,1-y_{i}({\vec {w}}\cdot {\vec {x}}_{i}-b)\right).$

$\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}({\vec {w}}\cdot {\vec {x}}_{i}-b)\right)\right]+\lambda \lVert {\vec {w}}\rVert ^{2},$

$\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}(w\cdot x_{i}-b)\right)\right]+\lambda \lVert w\rVert ^{2}.\qquad (2)$

${\vec {w}}=\sum _{i=1}^{n}c_{i}y_{i}{\vec {x}}_{i}$ .

$y_{i}({\vec {w}}\cdot {\vec {x}}_{i}-b)=1\iff b={\vec {w}}\cdot {\vec {x}}_{i}-y_{i}.$

${\vec {w}}=\sum _{i=1}^{n}c_{i}y_{i}\varphi ({\vec {x}}_{i}),$

${\begin{aligned}b={\vec {w}}\cdot \varphi ({\vec {x}}_{i})-y_{i}&=\left[\sum _{j=1}^{n}c_{j}y_{j}\varphi ({\vec {x}}_{j})\cdot \varphi ({\vec {x}}_{i})\right]-y_{i}\\&=\left[\sum _{j=1}^{n}c_{j}y_{j}k({\vec {x}}_{j},{\vec {x}}_{i})\right]-y_{i}.\end{aligned}}$

${\vec {z}}\mapsto \operatorname {sgn}({\vec {w}}\cdot \varphi ({\vec {z}})-b)=\operatorname {sgn} \left(\left[\sum _{i=1}^{n}c_{i}y_{i}k({\vec {x}}_{i},{\vec {z}})\right]-b\right).$

$f({\vec {w}},b)=\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}({\vec {w}}\cdot {\vec {x}}_{i}-b)\right)\right]+\lambda \lVert {\vec {w}}\rVert ^{2}.$

${\text{maximize}}\,\,f(c_{1}\ldots c_{n})=\sum _{i=1}^{n}c_{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}c_{i}(x_{i}\cdot x_{j})y_{j}c_{j},$

${\text{subject to }}\sum _{i=1}^{n}c_{i}y_{i}=0,\,{\text{and }}0\leq c_{i}\leq {\frac {1}{2n\lambda }}\;{\text{for all }}i.$

$\varepsilon (f)=\mathbb {E} \left[\ell (y_{n+1},f(X_{n+1}))\right].$

${\hat {\varepsilon }}(f)={\frac {1}{n}}\sum _{k=1}^{n}\ell (y_{k},f(X_{k})).$

$\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}(w\cdot x_{i}-b)\right)\right]+\lambda \lVert w\rVert ^{2}.$

[[Κατηγορία:Στατιστική ταξινόμηση]]