  We propose a nested reduced-rank regression (NRRR) approach in fitting
regression model with multivariate functional responses and predictors, to
achieve tailored dimension reduction and facilitate
interpretation/visualization of the resulting functional model. Our approach is
based on a two-level low-rank structure imposed on the functional regression
surfaces. A global low-rank structure identifies a small set of latent
principal functional responses and predictors that drives the underlying
regression association. A local low-rank structure then controls the complexity
and smoothness of the association between the principal functional responses
and predictors. Through a basis expansion approach, the functional problem
boils down to an interesting integrated matrix approximation task, where the
blocks or submatrices of an integrated low-rank matrix share some common row
space and/or column space. An iterative algorithm with convergence guarantee is
developed. We establish the consistency of NRRR and also show through
non-asymptotic analysis that it can achieve at least a comparable error rate to
that of the reduced-rank regression. Simulation studies demonstrate the
effectiveness of NRRR. We apply NRRR in an electricity demand problem, to
relate the trajectories of the daily electricity consumption with those of the
daily temperatures.
