  We propose a novel use of a broadcasting operation, which distributes
univariate functions to all entries of the tensor covariate, to model the
nonlinearity in tensor regression nonparametrically. A penalized estimation and
the corresponding algorithm are proposed. Our theoretical investigation, which
allows the dimensions of the tensor covariate to diverge, indicates that the
proposed estimation yields a desirable convergence rate. We also provide a
minimax lower bound, which characterizes the optimality of the proposed
estimator for a wide range of scenarios. Numerical experiments are conducted to
confirm the theoretical findings, and they show that the proposed model has
advantages over its existing linear counterparts.
