Learning with Transformation Invariant Kernels

This paper considers kernels invariant to translation, rotation and dilation. We show that no non-trivial positive definite (p.d.) kernels exist which are radial and dilation invariant, only conditionally positive definite (c.p.d.) ones. Accordingly, we discuss the c.p.d. On the practical side, we give a support vector machine (s.v.m.) algorithm for arbitrary c.p.d. For the thin- plate kernel this leads to a classifier with only one parameter (the amount of regu- larisation), which we demonstrate to be as effective as an s.v.m. with the Gaussian kernel, even though the Gaussian involves a second parameter (the length scale).