What is the proper way to extend kernel-based classifier to non-euclidean space like SO3? This kind of situation happens a lot in robotics, where the data points all live in a specific manifold. (Note: I don't mean manifold-learning, because we already know on what manifold the data lives.) As for kernel-based classifiers, I'm mainly concerning SVM and GP-classifier.
Say I'm using a radial-basis function (RBF) kernel, a naive idea arose to me is replacing the radius computation on Euclidean space by on rotational space. This seems to be able, for example, by using distance between two quaternion as described:
My question is following: For classification on a manifold, is it enough to just replace Eucledean metric by a proper metric suited for the specific manifold? If it is not enough, what kind of modification is needed?
It is nicer if you provide an academic paper describing the idea.