  Unlabeled data is a key component of modern machine learning. In general, the
role of unlabeled data is to impose a form of smoothness, usually from the
similarity information encoded in a base kernel, such as the
$\epsilon$-neighbor kernel or the adjacency matrix of a graph. This work
revisits the classical idea of spectrally transformed kernel regression (STKR),
and provides a new class of general and scalable STKR estimators able to
leverage unlabeled data. Intuitively, via spectral transformation, STKR
exploits the data distribution for which unlabeled data can provide additional
information. First, we show that STKR is a principled and general approach, by
characterizing a universal type of "target smoothness", and proving that any
sufficiently smooth function can be learned by STKR. Second, we provide
scalable STKR implementations for the inductive setting and a general
transformation function, while prior work is mostly limited to the transductive
setting. Third, we derive statistical guarantees for two scenarios: STKR with a
known polynomial transformation, and STKR with kernel PCA when the
transformation is unknown. Overall, we believe that this work helps deepen our
understanding of how to work with unlabeled data, and its generality makes it
easier to inspire new methods.
