  We introduce a new regression framework designed to deal with large-scale,
complex data that lies around a low-dimensional manifold with noises. Our
approach first constructs a graph representation, referred to as the skeleton,
to capture the underlying geometric structure. We then define metrics on the
skeleton graph and apply nonparametric regression techniques, along with
feature transformations based on the graph, to estimate the regression
function. We also discuss the limitations of some nonparametric regressors with
respect to the general metric space such as the skeleton graph. The proposed
regression framework suggests a novel way to deal with data with underlying
geometric structures and provides additional advantages in handling the union
of multiple manifolds, additive noises, and noisy observations. We provide
statistical guarantees for the proposed method and demonstrate its
effectiveness through simulations and real data examples.
