【ML算法学习】核K均值聚类Kernel K

法1：取特征空间 H = R 3 \mathcal{H}=R^3 H=R3，记 x = ( x ( 1 ) , x ( 2 ) ) T x=\left(x^{(1)}, x^{(2)}\right)^{\mathrm{T}} x=(x(1),x(2))T， z = ( z ( 1 ) , z ( 2 ) ) T z=\left(z^{(1)}, z^{(2)}\right)^{\mathrm{T}} z=(z(1),z(2))T，核函数为 ( x ⋅ z ) 2 = ( x ( 1 ) z ( 1 ) + x ( 2 ) z ( 2 ) ) 2 = ( x ( 1 ) z ( 1 ) ) 2 + 2 x ( 1 ) z ( 1 ) x ( 2 ) z ( 2 ) + ( x ( 2 ) z ( 2 ) ) 2 (x \cdot z)^2=\left(x^{(1)} z^{(1)}+x^{(2)} z^{(2)}\right)^2=\left(x^{(1)} z^{(1)}\right)^2+2 x^{(1)} z^{(1)} x^{(2)} z^{(2)}+\left(x^{(2)} z^{(2)}\right)^2 (x⋅z)2=(x(1)z(1)+x(2)z(2))2=(x(1)z(1))2+2x(1)z(1)x(2)z(2)+(x(2)z(2))2

所以可以取映射 ϕ ( x ) = ( ( x ( 1 ) ) 2 , 2 x ( 1 ) x ( 2 ) , ( x ( 2 ) ) 2 ) T \phi(x)=\left(\left(x^{(1)}\right)^2, \sqrt{2} x^{(1)} x^{(2)},\left(x^{(2)}\right)^2\right)^{\mathrm{T}} ϕ(x)=((x(1))2,2 ​x(1)x(2),(x(2))2)T，易验证 ϕ ( x ) ⋅ ϕ ( z ) = ( x ⋅ z ) 2 = K ( x , z ) \phi(x) \cdot \phi(z)=(x \cdot z)^2=K(x, z) ϕ(x)⋅ϕ(z)=(x⋅z)2=K(x,z)。

法2：取 H = R 3 \mathcal{H}=R^3 H=R3以及 ϕ ( x ) = 1 2 ( ( x ( 1 ) ) 2 − ( x ( 2 ) ) 2 , 2 x ( 1 ) x ( 2 ) , ( x ( 1 ) ) 2 + ( x ( 2 ) ) 2 ) T \phi(x)=\frac{1}{\sqrt{2}}\left(\left(x^{(1)}\right)^2-\left(x^{(2)}\right)^2, 2 x^{(1)} x^{(2)},\left(x^{(1)}\right)^2+\left(x^{(2)}\right)^2\right)^{\mathrm{T}} ϕ(x)=2 ​1​((x(1))2−(x(2))2,2x(1)x(2),(x(1))2+(x(2))2)T同样满足条件。

法3：取 H = R 4 \mathcal{H}=R^4 H=R4以及 ϕ ( x ) = ( ( x ( 1 ) ) 2 , x ( 1 ) x ( 2 ) , x ( 1 ) x ( 2 ) , ( x ( 2 ) ) 2 ) T \phi(x)=\left(\left(x^{(1)}\right)^2, x^{(1)} x^{(2)}, x^{(1)} x^{(2)},\left(x^{(2)}\right)^2\right)^T ϕ(x)=((x(1))2,x(1)x(2),x(1)x(2),(x(2))2)T满足条件。