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Abstract:
Abstract. 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 elementary
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 regularisation), 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).
