Skeleton Regression: A Graph-Based Approach to Estimation with Manifold Structure

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.