Does radial basis function kernel have a coefficient?

Question

I found there are two forms of RBF function.

1. There is a coefficient before $\exp$ $$ k_{f}\left(x_{i}, x_{j}\right)=\sigma^{2} \exp \left(-\frac{1}{2 \ell^{2}} \sum_{j=1}^{q}\left(x_{i, j}-x_{k, j}\right)^{2}\right) $$ which can be found in :

   Kernels in Gaussian Processes

   https://nipunbatra.github.io/blog/ml/2020/06/26/gp-understand.html#:~:text=The%20most%20commonly%20used%20kernel,exponential%20kernel%20%E2%80%93%20all%20are%20equivalent.&text=It%20has%20two%20parameters%2C%20described,2%20and%20the%20lengthscale%20l.&text=rbf.

2. There is no coefficient before $\exp$ $$ K\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\exp \left(-\frac{\left\|\mathbf{x} \quad \mathbf{x}^{\prime}\right\|^{2}}{2 \sigma^{2}}\right) $$ which can be found in wikipedia and scikit-learn:

   https://en.wikipedia.org/wiki/Radial_basis_function_kernel

   https://scikit-learn.org/stable/modules/gaussian_process.html#gp-kernels

What’s the difference between the two forms? Which is correct?

Answer 1

They are equivalent. The point of basis functions is to be able to use weighted linear combinations of them to approximate other functions, and the weighted linear combinations of these two give exactly the same set of approximations.