  Spherical regression explores relationships between variables on spherical
domains. We develop a nonparametric model that uses a diffeomorphic map from a
sphere to itself. The restriction of this mapping to diffeomorphisms is natural
in several settings. The model is estimated in a penalized maximum-likelihood
framework using gradient-based optimization. Towards that goal, we specify a
first-order roughness penalty using the Jacobian of diffeomorphisms. We compare
the prediction performance of the proposed model with state-of-the-art methods
using simulated and real data involving cloud deformations, wind directions,
and vector-cardiograms. This model is found to outperform others in capturing
relationships between spherical variables.
