Intrinsic Riemannian Functional Sufficient Dimension Reduction and Beyond

This paper focuses on linear sufficient dimension reduction with Riemannian random processes as predictors and complex random objects in a metric space as responses. We propose two novel methods—Intrinsic Riemannian Functional Weighted Inverse Regression Ensemble (iRF-WIRE) and Intrinsic Riemannian Functional Weighted Directional Regression (iRF-WDR)—to recover the central subspace. These methods can be readily extended to Wasserstein functional predictors. We establish their theoretical properties, including unbiasedness and optimal convergence rates, and conduct extensive simulation studies to assess their performance. Finally, we demonstrate the broad applicability of the proposed methods through two real-world datasets involving spherical functional data and Wasserstein functional data.