What will be the input value (i.e. $x$ and $x′$) of RBF kernel for a given dataset or data matrix $x$?

Question

If $x$ is a data matrix or dataset then What will be the input value (i.e. $x$ and $x'$) of RBF kernel $K_r(x,x')=\exp(-\frac{\|x-x'\|^2}{r})$ ?

I can understand $x$ is same as dataset or data matrix but what will be the value of $x'$ ?

At some place, people have taken $x$ is similar as dataset and $x'$ as transpose of $x$ but I didn't get what is the logic behind that?

PS: Just think above terminology as we use in MATLAB, because $x$ is generally a vector but we can do this type of calculation as a matrix in MATLAB.

Answer 1

Your notation is overloaded. Define $x$ as a vector containing data about a single observation, not a matrix and define $x'$ as a vector, possibly the same as $x$, possibly not. (And, if you need to refer to a matrix, one convention is to use a capital letter, such as $X$.)

Suppose you have two data points [1,0,3] and [1,2,3]. You have two data points, so the kernel matrix is 2 by 2. The diagonal elements must be 1 because the norm of a vector with itself is 0, and $\exp(0)=1$. The off-diagonal elements of the kernel is computed as $\exp(-\frac{||\langle 1,0,3\rangle-\langle 1,2,3\rangle||_2^2}{r})=\exp(-4/r)$. All Mercer kernels are symmetric, so element 1,2 is the same as element 2,1.