I have been searching the literature for a while on this and cannot find a clear explanation.
This relates to Radial Basis Function Interpolation but I think it applies more generally. I have a square matrix $r$ of interpoint distances for a sample. In order to solve the RBF equations, you apply a kernel and then invert.
Common kernels include $r$, $r^3$, or $r^2\log{r}$ (the latter is the "Thin Plate Spline". It is worth noting that $r\ge 0$. So here is my question: why not $r^2$? Or any one even power?
To be more clear, I understand that the resulting matrix is ill-conditioned but I do not understand from the mathematical standpoint what makes it so? Especially since $r\ge0$.
Can someone explain this to me?
Thanks!