  In the multivariate regression, also referred to as multi-task learning in
machine learning, the goal is to recover a vector-valued function based on
noisy observations. The vector-valued function is often assumed to be of low
rank. Although the multivariate linear regression is extensively studied in the
literature, a theoretical study on the multivariate nonlinear regression is
lacking. In this paper, we study reduced rank multivariate kernel ridge
regression, proposed by \cite{mukherjee2011reduced}. We prove the consistency
of the function predictor and provide the convergence rate. An algorithm based
on nuclear norm relaxation is proposed. A few numerical examples are presented
to show the smaller mean squared prediction error comparing with the
elementwise univariate kernel ridge regression.
