机器学习核函数手册

线性核:k(x, y) = x^T y + c

多项式核:k(x, y) = (alpha x^T y + c)^d

高斯核:k(x, y) = expleft(-frac{ lVert x-y rVert ^2}{2sigma^2}right) k(x, y) = expleft(- gamma lVert x-y rVert ^2 )

指数核:k(x, y) = expleft(-frac{ lVert x-y rVert }{2sigma^2}right)

拉普拉斯核:k(x, y) = expleft(- frac{lVert x-y rVert }{sigma}right)

ANOVA核:k(x, y) =  sum_{k=1}^n  exp (-sigma (x^k - y^k)^2)^d

双曲正切(sigmoid)核:k(x, y) = tanh (alpha x^T y + c)

有理二次核:k(x, y) = 1 - frac{lVert x-y rVert^2}{lVert x-y rVert^2 + c} 当高斯核代价太大时,可替代高斯核

多二次核:非正定内核k(x, y) = sqrt{lVert x-y rVert^2 + c^2}

逆多二次核:k(x, y) = frac{1}{sqrt{lVert x-y rVert^2 + theta^2}}

圆形核:k(x, y) = frac{2}{pi} arccos ( - frac{ lVert x-y rVert}{sigma}) - frac{2}{pi} frac{ lVert x-y rVert}{sigma} sqrt{1 - left(frac{ lVert x-y rVert}{sigma} right)^2}       mbox{if}~ lVert x-y rVert < sigma mbox{, zero otherwise}

球形核:k(x, y) = 1 - frac{3}{2} frac{lVert x-y rVert}{sigma} + frac{1}{2} left( frac{ lVert x-y rVert}{sigma} right)^3 mbox{if}~ lVert x-y rVert < sigma mbox{, zero otherwise},定义在R*R*R

波核(对称半正定):k(x, y) = frac{theta}{lVert x-y rVert right} sin frac{lVert x-y rVert }{theta}

能量核:k(x,y) = - lVert x-y rVert ^d

对数核:k(x,y) = - log (lVert x-y rVert ^d + 1) 条件正定

样条核:k(x, y) = 1 + xy + xy~min(x,y) - frac{x+y}{2}~min(x,y)^2+frac{1}{3}min(x,y)^3或者k(x,y) = prod_{i=1}^d 1 + x_i y_i + x_i y_i min(x_i, y_i) - frac{x_i + y_i}{2} min(x_i,y_i)^2 + frac{min(x_i,y_i)^3}{3}x,y in R^d

B-样条核:k(x, y) = prod_{p=1}^d B_{2n+1}(x_p - y_p)mbox{where~} p in N mbox{~with~} B_{i+1} := B_i otimes  B_0.

其中B_n(x) = frac{1}{n!} sum_{k=0}^{n+1} binom{n+1}{k} (-1)^k (x + frac{n+1}{2} - k)^n_+x^d_+ = begin{cases} x^d, & mbox{if }x > 0 \  0, & mbox{otherwise} end{cases}

贝塞尔核:k(x, y) = frac{J_{v+1}( sigma lVert x-y rVert)}{ lVert x-y rVert ^ {-n(v+1)} } 或者k(x,x') = - Bessel_{(nu+1)}^n (sigma |x - x'|^2)

柯西核:k(x, y) = frac{1}{1 + frac{lVert x-y rVert^2}{sigma^2} },来自于柯西分布

chi-square核:k(x,y) = 1 - sum_{i=1}^n frac{(x_i-y_i)^2}{frac{1}{2}(x_i+y_i)}或者,适用于SVM

直方图相交核也被称为最小核,可用于图像分类:k(x,y) = sum_{i=1}^n min(x_i,y_i)

广义相交核是基于图像分类的相交核,适用于更大的各种环境:k(x,y) = sum_{i=1}^m min(|x_i|^alpha,|y_i|^beta)

广义的T-student的核是一个mercel核,具有半正定矩阵:k(x,y) = frac{1}{1 + lVert x-y rVert ^d}


bayes核:k(x,y) = prod_{l=1}^N kappa_l (x_l,y_l),其中kappa_l(a,b) = sum_{c in {0;1}} P(Y=c mid X_l=a) ~ P(Y=c mid X_l=b),这个最常见,应取决于被建模的问题

小波核:k(x,y) = prod_{i=1}^N h(frac{x_i-c_i}{a}) :  h(frac{y_i-c_i}{a})

平移不变的核k(x,y) = prod_{i=1}^N h(frac{x_i-y_i}{a})

可能的H(x):h(x) = cos(1.75x)exp(-frac{x^2}{2}),小波核来自于小波分析

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