A1194
Title: Local Frechet regression with Riemannian manifold predictors
Authors: Chang Jun Im - Seoul National University (Korea, South) [presenting]
Jeong Min Jeon - Seoul National University (Korea, South)
Abstract: Recent advancements in object-oriented data analysis, particularly Frechet regression, have provided a robust framework for handling responses in general metric spaces. However, the existing literature predominantly assumes that predictors are Euclidean. A novel regression framework is proposed that accommodates responses in a general metric space and predictors residing on a Riemannian manifold. Intrinsic local constant and local linear estimators are developed that completely respect the underlying geometry of the manifold predictor space. By mapping the data to the tangent space via the logarithmic map and utilizing the volume density function, the theoretical properties of the proposed estimators are comprehensively established. Specifically, both pointwise and uniform convergence rates are derived under mild geometric regularity conditions. Notably, it is demonstrated that the estimators achieve the d-dimensional minimax optimal rate (where d is the intrinsic dimension of the manifold), exactly recovering the standard Euclidean rate when the curvature of the margin condition is quadratic.
