In the following paper:
<https://perso.crans.org/besson/publis/mva-2016/MVA_2015-16__Kernel_Methods__Homework__Besson_Clement_Zerbib.en.pdf>


problem 2, Kernel 9. it is shown that $K(x,y) = \frac{\min(x,y)}{\max(x,y)}$ is a positive definite kernel.


I am asking myself, if there is a feature mapping $\phi$ in a Hilbert space such that


$$\langle \phi(x) , \phi(y) \rangle = K(x,y)^2 = k(x,y)$$


"Motivation":
The Lorentz factor in special relativity has the form:


$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$


If we let $k(v\_1,v\_2) = (\frac{\min(v\_1,v\_2)}{\max(v\_1,v\_2)})^2$ then it is a product of two positive kernels, hence a p.d.k and the Euclidean distance between the two points in Hilbert space is:


$$d(v\_1,v\_2) := \sqrt{k(v\_1,v\_1)+k(v\_2,v\_2) - 2k(v\_1,v\_2)} = \sqrt{1+1-2\frac{v\_1^2}{v\_2^2}} = \sqrt{2}\sqrt{1-\frac{v\_1^2}{v\_2^2}}$$


so the Lorentz factor might be written as:


$$\gamma = \frac{\sqrt{2}}{d(v,c)}$$


where it would be interesting to interpret this further as the reciprocal of the distance between two points in some Hilbert space.


Also note the interesting fact, that:


$$k(v,v)= 1$$


which means that each $v$ lies in unit sphere of some Hilbert space.


(In QM those vectors are associated with pure states.)


(Also asked at MSE in case it is not research related: <https://math.stackexchange.com/questions/4181618/is-there-a-feature-mapping-for-this-kernel-kx-y-frac-minx-y-maxx-y> )



Is there a feature mapping for this kernel $k(x,y) = (\frac{\min(x,y)}{\max(x,y)})^2$?
The native Hilbert-space of $K^2$ is well known.


I assume the domain of $K$ is $\mathbb{R}^{>0}\times\mathbb{R}^{>0}.$ Note that:
$$ K(x,y)=\begin{cases}
\frac{x}{y} \text{ for } x\leq y\\
\frac{y}{x} \text{ for } x\geq y.\\
\end{cases}$$
Set $x = \exp u$ and $y = \exp v$ to obtain
$$ K(x,y)^2=K(\exp 2u, \exp 2v)=\exp\big(-2\lvert u-v\rvert\big)=G\_2(u,v).$$


This means $K^2$ is the pullback kernel of $G\_2$ by the pullback map $\log.$ For this pullback it is true (see for example [V. Paulsen, M. Raghupathi](https://www.cambridge.org/core/books/an-introduction-to-the-theory-of-reproducing-kernel-hilbert-spaces/C3FD9DF5F5C21693DD4ED812B531269A) Theorem 5.7) that


* $f\in\mathcal{H}(K^2)$ if and only if $f=g\circ\log$ for $g\in\mathcal{H}(G\_2)$
* and $\lVert f\rVert\_{K^2}=\lVert g\rVert\_{G\_2}.$


This specifies $\mathcal{H}(K^2)$ completely in terms of $\mathcal{H}(G\_2).$ The space $\mathcal{H}(G\_2)$ is equal as a set of functions to the Sobolev Space $W^{1,2}(\mathbb{R})$. It's norm is
$$ \lVert g\rVert^2\_{G\_2}=4 \left[\int\_\mathbb{R}g^2(u)\,du+4\int\_\mathbb{R}(g'(u))^2\,du\right].$$
(see [S. Saitoh, Y.Sawano](https://www.springer.com/gp/book/9789811005299) Theorem 1.7)


