Most regularized tensor regression research focuses on tensors predictors with scalars responses or vectors predictors to tensors responses. We consider the sparse low rank tensor on tensor regression where predictors $\mathcal{X}$ and responses $\mathcal{Y}$ are both high-dimensional tensors. By demonstrating that the general inner product or the contracted product on a unit rank tensor can be decomposed into standard inner products and outer products, the problem can be simply transformed into a tensor to scalar regression followed by a tensor decomposition. So we propose a fast solution based on stagewise search composed by contraction part and generation part which are optimized alternatively. We successfully demonstrate our method can out perform current methods in terms of accuracy and predictors selection by effectively incorporating the structural information.

翻译：多数常规的抗拉回归研究都侧重于具有卡路里反应或矢量预测的抗拉预测器的抗拉预测器。 我们考虑的是,在抗拉回归中,低级的低级抗拉是低级的。 预测器$\ mathcal{X}$和响应器$\ mathcal{Y}$都是高维的抗拉。 我们成功地展示了我们的方法可以通过有效地整合结构信息,在精度和预测器的选择方面完成目前的方法。