核函数kernel function汇总

1. 多项式核函数

2. 高斯核函数

RBF(radial basis function) 函数 $$ k(x, x') = exp(-\frac{1}{h} || x - x'||^2) $$ 对应 gradient: $$ \nabla_xk(x, x') = \sum \frac{2}{h} (x' - x)k(x, x') $$

1. KL-div 核函数 $$ k(x, x') = k(P(x) - P(x')) = P(x) log{\frac{P(x)}{P(x')}} $$ 对应 gradient: $$ \begin{aligned} \nabla_x k(x, x') = & \nabla(P(x)\log P(x) - P(x)\log P(x')) \ = & \nabla(P(x) \log P(x)) - \log P(x') \nabla P(x) \ = & \log P(x) \nabla P(x) + P(x) \nabla \log P(x) - \log P(x') \nabla P(x) \ = & P(x) \log P(x) \nabla \log P(x) + P(x) \nabla \log P(x) - P(x) \log P(x') \nabla \log P(x) \ = & (\log P(x) + 1 - \log P(x')) P(x) \nabla \log P(x) \ = & (P(x) \log \frac{P(x)}{P(x')} + P(x)) \nabla \log P(x) \ = & (k(x, x') + P(x) ) \nabla \log P(x) \end{aligned} $$

2. RBF(radial basis function) 函数 $$ x = \theta \ k(x, x') = exp(-\frac{1}{h} || x - x'||^2) $$ 对应 gradient: $$ \nabla_xk(x, x') = \sum \frac{2}{h} (x' - x)k(x, x') $$

3. RBF(radial basis function) 函数

$$ x = \theta \ k(x, x') = k(P(x), P(x')) = exp(-\frac{1}{h} || P(x) - P(x')||^2) $$

对应 gradient: $$ \nabla_xk(x, x') = \sum \frac{2}{h} (\nabla P(x') - \nabla P(x)) k(P(x), P(x')) $$