The dataset is generated from the tensor response regression (TRR) model:
$$Y_i = B X_i + \epsilon_i, i = 1,\ldots, n,$$
where \(n=20\) and the regression coefficient \(B \in R^{64\times 64}\) is a given image with rank 14, representing the mean difference of the response \(Y\) between two groups. To make the model conform to the envelope structure, we construct the envelope basis \(\Gamma_k\) and the covariance matrices \(\Sigma_k, k=1,2\), of error term as following. With the singular value decomposition of \(B\), namely \(B = \Gamma_1 \Lambda \Gamma_2^T\), we choose the envelope basis as \(\Gamma_k \in R^{64\times 14}, k=1,2\). Then the envelope dimensions are \(u_1 = u_2 = 14\). We generate another two matrices \(\Omega_k \in R^{14\times 14} = A_k A_k^T\) and \(\Omega_{0k} \in R^{50\times 50} = A_{0k}A_{0k}^T\), where \(A_k \in R^{14\times 14}\) and \(A_{0k} \in R^{50\times 50}\) are randomly generated from Uniform(0,1) elementwise. Then we set the covariance matrices \(\Sigma_k = \Gamma_k\Omega_k \Gamma_k^T + \Gamma_{0k}\Omega_{0k} \Gamma_{0k}^T\), followed by normalization with their Frobenius norms. We set the first 10 predictors \(X_i, i=1,\ldots, 10,\) as 1 and the rest as 0. The error term is then generated from two-way tensor (matrix) normal distribution \(TN( 0; \Sigma_1, \Sigma_2)\).