Additive models for symmetric positive-definite matrices and Lie groups

• 成果类型: Article
• 署名作者: Lin, Z.; Muller, H. G.; Park, B. U.
• 署名单位: National University of Singapore; University of California System; University of California Davis; Seoul National University (SNU)
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asac055
• 发表日期: 2023
• 页码: 361379
• 关键词: Principal component analysis; RIEMANNIAN-MANIFOLDS; Nonparametric Regression; Covariance matrices; Fréchet Regression; alzheimers-disease; functional data; interpolation; statistics; FRAMEWORK

摘要

We propose and investigate an additive regression model for symmetric positive-definite matrix-valued responses and multiple scalar predictors. The model exploits the Abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general Abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions, but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established, and numerical studies show that the proposed model enjoys good numerical performance, and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.