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Refinement of reproducing kernels. (English) Zbl 1235.68211
Summary: We continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel Hilbert space associated with the refinement kernel contains that with the original kernel as a subspace. To motivate this study, we first develop a refinement kernel method for learning, which gives an efficient algorithm for updating a learning predictor. Several characterizations of refinement kernels are then presented. It is shown that a nontrivial refinement kernel for a given kernel always exists if the input space has an infinite cardinal number. Refinement kernels for translation invariant kernels and Hilbert-Schmidt kernels are investigated. Various concrete examples are provided.

MSC:
68T05 Learning and adaptive systems in artificial intelligence
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
68Q32 Computational learning theory
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