Tensor train regression with convex regularization

Tensor regression has received increasing attention in recent years since more and more data sets are naturally represented in tensor structures. In the regression setting, most existing works are concerned with tensors in CP or Tucker formats. In particular, the statistical rates for regression problems with a tensor coefficient that has a low-rank tensor train (TT) format have not been established. In this paper, we study tensor train regression (with tensor predictors and scalar responses) using convex regularization. The statistical rates of the estimators are established in the high-dimensional setting, for both mean regression and quantile regression. Some numerical experiments and empirical application are presented to show their finite-sample performance. This work provides theoretical guarantees for a stable and efficient alternative for high-dimensional tensor regression based on TT format, which scales well for high-order tensors.