Extrinsic Local Regression on Manifold-Valued Data
提出一种外源回归框架，用于处理响应变量位于流形上、预测变量为欧几里得数据的回归问题，通过嵌入高维空间进行局部回归再投影回流形，适用于形状分析、神经科学等领域。
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 i.i.d 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 are considered indicating the wide applicability of our approach.