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

$$<\phi(x) , \phi(y)> = K(x,y)^2 = k(x,y)$$

"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 Hilber 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 pures 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 )

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 )

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

$$<\phi(x) , \phi(y)> = K(x,y)^2 = k(x,y)$$

"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 Hilber space.

(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 )

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

$$<\phi(x) , \phi(y)> = K(x,y)^2 = k(x,y)$$

"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 Hilber 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 pures 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 )