In practice, covariance matrices are specified using functions known as kernels. You may find more than one definition of kernel in the statistical literature, with slightly different mathematical properties. For the purpose of our discussion, we are going to say that a kernel is basically a symmetric function that takes two inputs and returns a value of zero in the inputs are the same or positive otherwise. If these conditions are met, we can interpret the output of a kernel function as a measure of similarity between the two inputs.
Among the many useful kernels available, a popular one used is the exponentiated quadratic kernel:
Here, is the squared Euclidean distance:
It may not be obvious at first ...
