Local Fréchet Regression With Toroidal Predictors

We provide the first regression framework that simultaneously accommodates responses taking values in a general metric space and predictors lying on a general torus. We propose intrinsic local constant and local linear estimators that respect the underlying geometries of both the response and predictor spaces. Our local linear estimator differs from existing approaches even when the responses are scalar. For both proposed estimators, we establish consistency and convergence rates. Simulation studies with scalar and spherical responses, together with a real data application involving graph‐Laplacian‐valued responses, illustrate the practical advantages of the proposed methodology.