Decomposable Sparse Tensor on Tensor Regression
This work addresses tensor regression problems in high-dimensional data analysis, representing an incremental improvement over existing methods.
The paper tackles sparse low-rank tensor-on-tensor regression where both predictors and responses are high-dimensional tensors, proposing a fast stagewise search method that outperforms current approaches in accuracy and predictor selection by leveraging structural information.
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.