Programming Hints for Kernel–Based Methods

Hopefully helpful stuff for writing prototypes of programs for kernel–based methods, in particular for radial kernels or matrix–valued divergence–free kernels. The focus is on MATLAB programming. None of the programs is optimized for anything, in particular not for speed. 1 Basics We assume x, y to be vectors in R d , and we consider radial kernels of the form K(x, y) = φ(x − y 2), for all x, y ∈ R d. We store points in MATLAB/FORTRAN–style as rows of matrices with d For univariate cases, note that MATLAB sequences like t=-1:0.01:1 generate a row, not a column. Random sets of n points in d dimensions within [0, 1] d are generated via p=rand(n,d). To generate random points in [a, b] d , use p=a+(b-a)*rand(n,d). Regular grids are generated by the meshgrid command. The standard 2D case looks like [x y]=meshgrid(a:h:b,a:h:b); for generating points in [a, b] 2 with spacing h. But these are not points in our matrix convention. Both x and y are matrices of the same shape, with identical columns or rows. Use p=[x(:) y(:)]; to get a point matrix with two columns. The inverse operation is