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Learning the kernel function via regularization AbstractWe study the problem of finding an optimal kernel from a prescribed convex set of kernels $\calK$ for learning a real-valued function by regularization. We establish for a wide variety of regularization functionals that this leads to a convex optimization problem and, for square loss regularization, we characterize the solution of this problem. We show that, although $\calK$ may be an uncountable set, the optimal kernel is always obtained as a convex combination of at most $m+1$ kernels, where $m$ is the number of data examples. In particular, our results apply to learning the optimal radial kernel or the optimal dot product kernel.
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