What advantage do quadratic basis functions have over gaussian basis functions in RBF implementations?

Question

I've been reading up recently on radial basis functions, and I've recently moved from Gaussian basis functions of the form $\phi(x)=\exp(-\alpha||x-x_i||)$ to multiquadratic basis functions. My source is this paper, which is referenced in an algorithm that I'm looking at.

The basis function for the multiquadratic RBF is given as $\phi(x)=\sqrt{||x-x_i||^2+R^2}$, where $R$ is a shape parameter. While it's certainly possible to solve the set of linear equations in a basis function and get a valid result for any equation, this choice of basis functions seems odd. Compared to a Gaussian basis function, where the effect of a basis function with a large value of $||x-x_i||$ approaches zero, multiquadratics grow increasingly rapidly as the radius increases.

Intuitively, this seems like a strange choice for a basis function, since distant points in a training data set would seemingly influence the results of a local solution more than nearby points. Why is this function used as a basis function, and what advantages does it have over something like a Gaussian basis function?

Answer 1

...a strange choice for a basis function, since distant points in a training data set would seemingly influence the results of a local solution more than nearby points. Why is this function used as a basis function?

The sum of their influences quickly fades away. It's well illustrated in 1d, where we can represent a spline either piece-wise, locally controlled, or as one, 'global' RBF sum with $|x|$ kernel (see here for example). This might seem strange as each $c_i|x-x_i|$ term is unbounded. Yet, together they cancel out automagically at a distance. You can see it by following the derivation at the link. Another example is a system of electric charges overall neutral.

what advantages does it have over something like a Gaussian basis function?

Multiquadric and other globally supported RBF have a good hole-filling property: if you have holes that are irregular with unknown scales, they're going to be smoothly interpolated over according to surrounding trends from neighborhoods of corresponding scales. If your dataset has a regular sparsity of one radius you can estimate, you can use instead compactly supported RBF, or Gaussian RBF which is practically local.