SVM 超平面计算例题

Posted 2022-12-07 ACxz

SVM Summary

Example

Suppose the dataset contains two positive samples $x^{(1)}=[1,1{]}^T$ and$x^{(2)}=[2,2{]}^T$, and two negative samples $x^{(3)}=[0,0{]}^T$ and $x^{(4)}=[-1,0{]}^T$. Please calculate the SVM decision hyperplane.

Calculate

$$\underset{\lambda }{\min }\mathcal{J}(\lambda )=\frac{1}{2}\sum _{i=1}^N\sum _{j=1}^N{\lambda }_i{\lambda }_j{y}^{(i)}{y}^{(j)}(x^{(i)}{)}^T{x}^{(j)}-\sum _{i=1}^N{\lambda }_i$$
$$s.t.{\lambda }_i⩾0,\sum _{i=1}^N{\lambda }_i{y}^{(i)}=0$$
由$DatasetD:x:[1,1],[2,2],[0,0],[-1,0],y:1,1,-1,-1$可得下式：
$$\begin{gathered} \underset{\lambda }{\min }\mathcal{J}(\lambda )=\frac{1}{2}(2{\lambda }_1^2+8{\lambda }_2^2+{\lambda }_4^2+8{\lambda }_1{\lambda }_2+2{\lambda }_1{\lambda }_4+4{\lambda }_2{\lambda }_4) \\ -{\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4 \\ s.t{\lambda }_1⩾0,{\lambda }_2⩾0,{\lambda }_3⩾0,{\lambda }_4⩾0 \\ {\lambda }_1+{\lambda }_2-{\lambda }_3-{\lambda }_4=0 \end{gathered}$$
since ${\lambda }_1+{\lambda }_2={\lambda }_3+{\lambda }_4\to {\lambda }_3={\lambda }_1+{\lambda }_2-{\lambda }_4$:
min ⁡ λ   J ( λ ) = λ 1 2 + 4 λ 2 2 + 1 2 λ 4 2 + 4 λ 1 λ 2 + λ 1 λ