Boosted Sparse and Low-Rank Tensor Regression
This work addresses interpretable modeling for tensor data in fields like genomics or imaging, but it is incremental as it builds on existing tensor regression methods.
The paper tackles the problem of relating a univariate outcome to a feature tensor by proposing a sparse and low-rank tensor regression model, which achieves superior performance on real-world and synthetic examples.
We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, for the unit-rank tensor regression, we propose a stagewise estimation procedure to efficiently trace out its entire solution path. We show that as the step size goes to zero, the stagewise solution paths converge exactly to those of the corresponding regularized regression. The superior performance of our approach is demonstrated on various real-world and synthetic examples.