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Keywords: nonparametric regression, manifold learning, kernel regression, spline
In this paper, we propose a novel regression framework that can deal with input data concentrated on a low-dimension manifold. The proposed framework first learns a graph representation, which we call the skeleton, of the input data to summarize the manifold structure. Then nonparametric regression methods are applied to estimate the regression function on the skeleton. We derive some associated statistical and computational theories and use simulation analysis to illustrate the accuracy of our methods. We also discussed the feasibility and the issue of edge directions when fitting splines of derivative smoothness on graphs. Our framework also has the advantage that predictors from distinct manifolds can be accounted for and is robust to additive noise and noisy observations.