We introduce a new convex optimization framework for logistic scalar-on-matrix regression which incorporates nuclear and $\ell_1$ norm penalties to enforce simultaneously low-rank and sparse structures in the estimated coefficient matrix. The proposed method enables interpretable modeling of high-dimensional matrix-valued predictors in the presence of binary responses. We derive a custom algorithm based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve the resulting convex optimization problem and establish the theoretical properties of the obtained solution. Numerical experiments clearly demonstrate the effectiveness of our method in recovering meaningful predictive patterns. Finally, we apply our method to the brain imaging data to identify structures in functional brain connectivity matrices that are characteristic of subjects with a family history of alcohol use disorders (AUDs).

翻译：我们提出了一种用于逻辑矩阵回归的新型凸优化框架，该框架通过引入核范数和$\ell_1$范数惩罚项，使得估计的系数矩阵同时具有低秩和稀疏结构。该方法能够在二值响应存在的情况下，对高维矩阵值预测变量进行可解释性建模。我们基于交替方向乘子法（ADMM）推导了一种定制算法，以高效求解所得到的凸优化问题，并建立了所得解的理论性质。数值实验清晰证明了本方法在恢复有意义预测模式方面的有效性。最后，我们将该方法应用于脑影像数据，用于识别在具有酒精使用障碍（AUDs）家族史的受试者中具有特征性的功能性脑连接矩阵结构。