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We consider the general problem of learning an unknown functional dependency, f : X->Y, between a structured input space X and a structured output space Y, from labeled and unlabeled examples. We formulate this problem in terms of data-dependent regularization in Vector-valued Reproducing Kernel Hilbert Spaces (Micchelli & Pontil, 2005) which elegantly extend familiar scalar-valued kernel methods to the general setting where Y has a Hilbert space structure. Our methods provide a natural extension of Manifold Regularization (Belkin et al., 2006) algorithms to also exploit output inter-dependencies while enforcing smoothness with respect to input data geometry. We propose a class of matrix-valued kernels which allow efficient implementations of our algorithms via the use of numerical solvers for Sylvester matrix equations. On multilabel image annotation and text classification problems, we find favorable empirical comparisons against several competing alternatives.
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