2007
NeurIPS
NeurIPS 2007
Learning with Transformation Invariant Kernels
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 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).
Authors
Topics
Machine Learning > Core Methods > Classification
Machine Learning > Core Methods > Representation Learning
Machine Learning > Optimization & Theory > Optimization
Machine Learning > Optimization & Theory > Statistical Learning
Machine Learning > Core Methods > Kernel Methods
Machine Learning > Core Methods > Support Vector Machine
Keywords
representer theorem
pattern classification
transformation invariant kernels
conditionally positive definite
transformation invariance
gaussian kernel
support vector machine
translation invariance
positive definite kernel
invariant kernel
kernel methods
dilation invariance
conditionally positive definite kernel
thin-plate kernel