Intrinsic Functional Sliced Inverse Regression on Riemannian Manifolds and Wasserstein Spaces

By defining a logarithmic functional central subspace along the Frchet mean curve, a novel intrinsic functional sliced inverse regression model is proposed for a Riemannian or Wasserstein functional predictor and a scalar response.This approach is applicable to both Euclidean submanifolds and manifolds without a natural ambient space.Under mild conditions, a truncation-based estimation procedure is developed for the logarithmic functional central subspace, and its asymptotic properties are subsequently investigated within the framework of intrinsic geometry.Furthermore, the flatness of the Wasserstein space ensures that the proposed method can identify the optimal truncation number to achieve the minimax optimal convergence rate for estimating the logarithmic functional central subspace.The finite-sample performance of the proposed method is demonstrated through both simulation studies and real data applications.