Extrinsic Local Regression on Manifold-Valued Data

• 成果类型: Article
• 署名作者: Lin, Lizhen; St Thomas, Brian; Zhu, Hongtu; Dunson, David B.
• 署名单位: University of Texas System; University of Texas Austin; University of Notre Dame; Duke University; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2016.1208615
• 发表日期: 2017
• 页码: 1261-1273
• 关键词: large-sample theory; DENSITY-ESTIMATION; linear-regression; diffusion

摘要

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.