  Low-rank matrix decomposition has gained great popularity recently in scaling
up kernel methods to large amounts of data. However, some limitations could
prevent them from working effectively in certain domains. For example, many
existing approaches are intrinsically unsupervised, which does not incorporate
side information (e.g., class labels) to produce task specific decompositions;
also, they typically work "transductively", i.e., the factorization does not
generalize to new samples, so the complete factorization needs to be recomputed
when new samples become available. To solve these problems, in this paper we
propose an"inductive"-flavored method for low-rank kernel decomposition with
priors. We achieve this by generalizing the Nystr\"om method in a novel way. On
the one hand, our approach employs a highly flexible, nonparametric structure
that allows us to generalize the low-rank factors to arbitrarily new samples;
on the other hand, it has linear time and space complexities, which can be
orders of magnitudes faster than existing approaches and renders great
efficiency in learning a low-rank kernel decomposition. Empirical results
demonstrate the efficacy and efficiency of the proposed method.
