Multilinear Regression for Embedded Feature Selection with Application to fMRI Analysis

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 `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.