Does radial basis function kernel has a coefficient?

Question

I found there are two forms of RBF function.

1. these 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. these 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

Now the question is what’s the difference between the two forms? which is True ?

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.