We develop a Bayesian methodology aimed at simultaneously estimating low-rank and row-sparse matrices in a high-dimensional multiple-response linear regression model. We consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank, and shrinks the regression matrix towards low-rank and row-sparse structures. We provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra-high dimensional settings where the number of predictors can grow sub-exponentially relative to the sample size. A one-step post-processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection, with default choices of tuning parameters. We additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example.

中文翻译：

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贝叶斯稀疏多元回归用于同时降阶和变量选择

我们开发了一种贝叶斯方法，旨在同时估计高维多响应线性回归模型中的低秩矩阵和行稀疏矩阵。我们在回归系数矩阵上考虑了精心设计的收缩先验，这避免了在秩上指定先验的需要，并将回归矩阵向低秩和行稀疏结构收缩。我们通过在超高维设置中的预测风险下证明后验均值的极小极大优化，为所提出的方法提供理论支持，其中预测变量的数量可以相对于样本大小呈次指数增长。提出了一种由估计系数矩阵的行上的组套索惩罚引起的一步后处理方案，用于变量选择，默认选择调整参数。我们还使用一种新颖的优化函数来提供对秩的估计，以实现协变量空间的降维。我们在广泛的模拟研究和真实数据示例中展示了所提出方法的性能。