  This article aims to seek a selection and estimation procedure for a class of
tensor regression problems with multivariate covariates and matrix responses,
which can provide theoretical guarantees for model selection in finite samples.
Considering the frontal slice sparsity and low-rankness inherited in the
coefficient tensor, we formulate the regression procedure as a group SLOPE
penalized low-rank tensor optimization problem based on an orthogonal
decomposition, namely TgSLOPE. This procedure provably controls the newly
introduced tensor group false discovery rate (TgFDR), provided that the
predictor matrix is column-orthogonal. Moreover, we establish the
asymptotically minimax convergence with respect to the TgSLOPE estimate risk.
For efficient problem resolution, we equivalently transform the TgSLOPE problem
into a difference-of-convex (DC) program with the level-coercive objective
function. This allows us to solve the reformulation problem of TgSLOPE by an
efficient proximal DC algorithm (DCA) with global convergence. Numerical
studies conducted on synthetic data and a real human brain connection data
illustrate the efficacy of the proposed TgSLOPE estimation procedure.
