Optimal function-on-scalar regression over complex domains

In this work we consider the problem of estimating function-on-scalar regression models when functions are observed over multi-dimensional or manifold domains and with potentially multivariate output. We establish minimax rates convergence present an estimator based on reproducing kernel Hilbert spaces that achieves rate. To better interpret derived rates, extend well-known links between RKHS Sobolev to case where domain is a compact Riemannian manifold. This accomplished using interesting connection Weyl’s Law from partial differential equations. conclude numerical study application 3D facial imaging.