Tensor sliced inverse regression

Sliced inverse regression (SIR) is a widely used non-parametric method for supervised dimension reduction. Conventional SIR mainly tackles simple data structure but is inappropriate for data with array (tensor)-valued predictors. Such data are commonly encountered in modern biomedical imaging and social network areas. For these complex data, dimension reduction is generally demanding to extract useful information from abundant measurements. In this article, we propose higher-order sufficient dimension reduction mainly by extending SIR to general tensor-valued predictors and refer to it as tensor SIR. Tensor SIR is constructed based on tensor decompositions to reduce a tensor-valued predictor's multiple dimensions simultaneously. The proposed method provides fast and efficient estimation. It circumvents high-dimensional covariance matrix inversion that researchers often suffer when dealing with such data. We further investigate its asymptotic properties and show its advantages by simulation studies and a real data application.