and on the square [- ] quasi-random point sets have better space-filling properties than either regular grid of points or set of pseudo-random points we refer to [ for details note that there is no observation noise in this particular problem - - - figure peaks function sampled at hammersley points the purpose of this example is to illustrate howusing the small data set of size the entire peaks function can be approximated well using kernel methods in particularwe use the gaussian kernel ( on and denote by the unique rkhs corresponding to this kernel we omit the regularization term in ( )and thus our objective is to find the solution to min (yi (xi )) gh = by the representer theoremthe optimal function is of the form kx xi (xai exp = where :[ an ]isby ( )the solution to the set of linear equations kka ky note that we are performing regression over the class of functions with an implicit feature space due to the representer theoremthe solution to this problem coincides with the solution to the linear regression problem for which the -th feature (for nis chosen to be the vector [ ( xi ) (xn xi )]the following code performs these calculations and gives the contour plots of and the peaks functionsshown in figure we see that the two are quite close code for the generation of hammersley points is available from the book' github site as genham py peakskernel py from genham import hammersley import numpy as np import matplotlib pyplot as plt from mpl_toolkits mplot import axes