Abstract: Sufficient dimension reduction methods are exible tools for data visualization and exploratory analysis, typically in a regression of a univariate response on a multivariate predictor. Recently, there has been growing interest in the analysis of matrix-variate and tensor-variate data. For regressions with tensor predictors, a general framework of dimension folding and several moment-based estimation procedures have been proposed in the literature. In this article, we propose two likelihood-based dimension folding methods motivated by quadratic discriminant analysis for tensor data: the maximum likelihood estimators are derived under a general covariance setting and a structured envelope covariance setting. We study the asymptotic properties of both estimators and show using simulation studies and a real-data analysis that they are more accurate than existing moment-based estimators.

Key words and phrases: Dimension folding, quadratic discriminant analysis, sufficient dimension reduction, tensor.