Minimax estimation in multi-task regression under low-rank structures

This study investigates the minimaxity of a multi-task nonparametric regression problem. We formulate a simultaneous function estimation problem based on information pooling across multiple experiments under a low-dimensional structure. A nonparametric reduced rank regression estimator based on the nuclear norm penalisation scheme is proposed to incorporate the low-dimensional structure in the estimation process. A rank of a set of functions is defined in terms of their Fourier coefficients to formally characterise the dependence structure among functions. Minimax upper and lower bounds are established under various asymptotic scenarios to examine the role of the low-rank structure in determining optimal rates of convergence. The results confirm that exploiting the low-rank structure can significantly improve the convergence rate for the simultaneous estimation of multiple functions. The results also imply that the proposed estimator is rate optimal in the minimax sense for the rank-constraint Sobolev class of vector-valued functions.