  We present a general framework to learn functions in tensor product
reproducing kernel Hilbert spaces (TP-RKHSs). The methodology is based on a
novel representer theorem suitable for existing as well as new spectral
penalties for tensors. When the functions in the TP-RKHS are defined on the
Cartesian product of finite discrete sets, in particular, our main problem
formulation admits as a special case existing tensor completion problems. Other
special cases include transfer learning with multimodal side information and
multilinear multitask learning. For the latter case, our kernel-based view is
instrumental to derive nonlinear extensions of existing model classes. We give
a novel algorithm and show in experiments the usefulness of the proposed
extensions.
