Motivated by applications in various scientific fields having demand of predicting relationship between higher-order (tensor) feature and univariate response, we propose a \underline{S}parse and \underline{L}ow-rank \underline{T}ensor \underline{R}egression model (SLTR). This model enforces sparsity and low-rankness of the tensor coefficient by directly applying $\ell_1$ norm and tensor nuclear norm on it respectively, such that (1) the structural information of tensor is preserved and (2) the data interpretation is convenient. To make the solving procedure scalable and efficient, SLTR makes use of the proximal gradient method to optimize two norm regularizers, which can be easily implemented parallelly. Additionally, a tighter convergence rate is proved over three-order tensor data. We evaluate SLTR on several simulated datasets and one fMRI dataset. Experiment results show that, compared with previous models, SLTR is able to obtain a solution no worse than others with much less time cost.

Embedded feature selection is effective when both prediction and interpretation are needed. The Lasso and its extensions are standard methods for selecting a subset of features while optimizing a prediction function. In this paper, we are interested in embedded feature selection for multidimensional data, wherein (1) there is no need to reshape the multidimensional data into vectors and (2) structural information from multiple dimensions are taken into account. Our main contribution is a new method called Regularized multilinear regression and selection (Remurs) for automatically selecting a subset of features while optimizing prediction for multidimensional data. Both nuclear norm and the ℓ 1 -norm are carefully incorporated to derive a multi-block optimization algorithm with proved convergence. In particular, Remurs is motivated by fMRI analysis where the data are multidimensional and it is important to find the connections of raw brain voxels with functional activities. Experiments on synthetic and real data show the advantages of Remurs compared to Lasso, Elastic Net, and their multilinear extensions.