Extrinsic Local Regression on Manifold-Valued Data

We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging, and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling iid manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient, and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples is considered indicating the wide applicability of our approach. Supplementary materials for this article are available online.