  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.
