Vector-valued manifold regularization

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 multi-label image annotation and text classification problems, we find favorable empirical comparisons against several competing alternatives. Copyright 2011 by the author(s)/owner(s).