The shape parameter in the Gaussian function
This is the fifth of our series of works about the shape parameter. We now explore the parameter ? ? contained in the famous Gaussian function e ? ? | x | 2 , x ? R n . In the theory of radial basis functions (RBFs), the Gaussian is frequently used in virtue of its good error bound and numerical tractability. However, the optimal choice of ? ? has been unknown. People conversant with RBFs know that ? ? is very influential, but do not have a reliable criterion of its choice. The purpose of this paper is to uncover its mystery. In particular, we have greatly improved the result of Madych (1992) in [15], and we present a concrete function of ? ? which shows the influence of ? ? in the error estimate of Gaussian interpolation and with which the optimal ? ? can always be found.
shape parameter gaussian radial basis function interpolation