Christian Walder, Olivier Chapelle
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. case and provide some novel analysis, including an elemen- tary derivation of a c.p.d. representer theorem. On the practical side, we give a support vector machine (s.v.m.) algorithm for arbitrary c.p.d. kernels. 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).