Denoting the scale and threshold parameters by $\gamma>0$ and $\theta$, in the available literature on SVM, the mapping of a kernel function of two vectors $x$ and $y$ into the reproducing Hilbert space is generally illustrated by two common examples, the polynomial kernel of integer degree $d$, defined by $(\gamma\langle x,y\rangle + \theta)^d$, and the Gaussian kernel, defined by $\exp(-\gamma||x-y||_2^2)$.
- In the case of a polynomial, the mapping is direct and expressed in terms of multinomial coefficients and degree combinations of $x$ and $y$.
- In the case of a Gaussian kernel, the mapping is achieved by using the infinite Taylor expansion of the term $\exp(2\gamma\langle x,y\rangle)$, after stripping out $\exp(-\gamma||x||_2^2)$ and $\exp(-\gamma||y||_2^2)$.
Do you know where I could find a solution or a method to obtain the reproducing mapping function in the case of the following kernels, whose Taylor expansions do not lead to an obvious decomposition between the terms in $x$ and in $y$, on the one hand, and the terms in $\langle x,y\rangle$, on the other hand:
- Sigmoid (hyperbolic tangent): $\tanh(\gamma\langle x,y\rangle + \theta)$
- Laplace: $\exp(-\gamma||x-y||_1)$
- Cauchy ($d=1$) or Student ($d>1$) : $\frac{1}{(1+\gamma||x-y||_2^2)^{\frac{d+1}{2}}}$