NONLINEAR GLOBAL FRÉCHET REGRESSION FOR RANDOM OBJECTS VIA WEAK CONDITIONAL EXPECTATION

• 成果类型: Article
• 署名作者: Bhattacharjee, Satarupa; Li, Bing; Xue, Lingzhou
• 署名单位: State University System of Florida; University of Florida; Florida State University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2457
• 发表日期: 2025
• 页码: 117-143
• 关键词: sufficient dimension reduction; extrinsic sample means; frechet means; EXISTENCE; Consistency; statistics; MANIFOLDS; SPACE

摘要

Random objects are complex non-Euclidean data taking values in general metric spaces, possibly devoid of any underlying vector space structure. Such data are becoming increasingly abundant with the rapid advancement in technology. Examples include probability distributions, positive semidefinite matrices and data on Riemannian manifolds. However, except for regression for object-valued response with Euclidean predictors and distributionon-distribution regression, there has been limited development of a general framework for object-valued response with object-valued predictors in the literature. To fill this gap, we introduce the notion of a weak conditional Fr & eacute;chet mean based on Carleman operators and then propose a global nonlinear Fr & eacute;chet regression model through the reproducing kernel Hilbert space (RKHS) embedding. Furthermore, we establish the relationships between the conditional Fr & eacute;chet mean and the weak conditional Fr & eacute;chet mean for both Euclidean and object-valued data. We also show that the state-of-the-art global Fr & eacute;chet regression developed by Petersen and M & uuml;ller (Ann. Statist. 47 (2019) 691-719) emerges as a special case of our method by choosing a linear kernel. We require that the metric space for the predictor admits a reproducing kernel, while the intrinsic geometry of the metric space for the response is utilized to study the asymptotic properties of the proposed estimates. Numerical studies, including extensive simulations and a real application, are conducted to investigate the finite-sample performance.