Kernel properties and feature maps

[akerman]

I am preparing myself for maths exam and I am really struggling with kernels.
I have following six kernels and I need to prove that each of them is valid and derive feature map.
1) K(x,y) = g(x)g(y), g:R^d -> R
With this one I know it is valid but I don't know how to prove it. Also is g(x) a correct feature map?

2) K(x,y) = x^T * D * y, D is diagonal matrix with no negative entries
With this one I am also sure that it is valid but I have no idea how to prove it or derive feature map

For the following four I don't know anything.
3) K(x,y) = x^T * y - (x^T * y)^2
4) K(x,y) =$\prod_{i=1}^{d} x_{i}y_{i}$
5) cos(angle(x,x'))
6) min(x,x'), x,x' >=0

Please help me as I am very struggling with kernel methods and if you could please provide as much explanation as possible

 

[jvicens]

as I am still learning.

Hello,

As a fellow scientist, I understand the struggle of preparing for exams and dealing with difficult concepts like kernels. I will do my best to explain each kernel and how they can be proven to be valid.

1) K(x,y) = g(x)g(y), g:R^d -> R
This kernel is known as the "polynomial kernel" and it is valid because it satisfies the definition of a kernel function, which is a symmetric positive semi-definite function. To prove this, we can use the Mercer's theorem, which states that any valid kernel function can be expressed as a dot product in a higher dimensional space. In this case, we can express K(x,y) as g(x)g(y) in a higher dimensional space, which means that g(x) is a valid feature map.

2) K(x,y) = x^T * D * y, D is diagonal matrix with no negative entries
This kernel is known as the "diagonal kernel" and it is valid because it can also be expressed as a dot product in a higher dimensional space. To prove this, we can use the same approach as in the previous kernel and express K(x,y) as a dot product in a higher dimensional space, where the diagonal matrix D is represented by the diagonal elements of the feature map.

3) K(x,y) = x^T * y - (x^T * y)^2
This kernel is known as the "sigmoid kernel" and it is valid because it can be expressed as a dot product in a higher dimensional space. To prove this, we can use the same approach as in the previous kernels and express K(x,y) as a dot product in a higher dimensional space, where the feature map is given by (x, x^2, y, y^2, xy, x^2y, xy^2, x^2y^2).

4) K(x,y) =$\prod_{i=1}^{d} x_{i}y_{i}$
This kernel is known as the "element-wise multiplication kernel" and it is valid because it can also be expressed as a dot product in a higher dimensional space. To prove this, we can again use the same approach and express K(x,y) as a dot product in a higher dimensional space, where the feature map is given by (x_1y_1, x_2y_2