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.

翻译：我们提出了一种广播操作的新颖应用，该操作将单变量函数分布到张量协变量的所有元素上，以非参数方式对张量回归中的非线性进行建模。我们提出了一种带罚项的估计方法及相应算法。我们的理论研究允许张量协变量的维度发散，表明所提出的估计方法具有理想的收敛速度。我们还提供了一个极小极大下界，该下界刻画了所提出的估计量在广泛场景下的最优性。通过数值实验验证了理论结果，并表明所提出的模型相较于现有线性模型具有优势。