  Recently, tensor data (or multidimensional array) have been generated in many
modern applications, such as functional magnetic resonance imaging (fMRI) in
neuroscience and videos in video analysis. Many efforts are made in recent
years to predict the relationship between tensor features and univariate
responses. However, previously proposed methods either lose structural
information within tensor data or have prohibitively expensive time costs,
especially for large-scale data with high-order structures. To address such
problems, we propose the Sparse and Low-rank Tensor Regression (SLTR) model.
Our model enforces sparsity and low-rankness of the tensor coefficient by
directly applying $\ell_1$ norm and tensor nuclear norm, such that it preserves
structural information of the tensor. To make the solving procedure scalable
and efficient, SLTR makes use of the proximal gradient method, which can be
easily implemented parallelly. We evaluate SLTR on several simulated datasets
and one video action recognition dataset. Experiment results show that,
compared with previous models, SLTR can obtain a better solution with much
fewer time costs. Moreover, our model's predictions exhibit meaningful
interpretations on the video dataset.
