Reduced-rank regression estimates regression coefficients by imposing a low-rank constraint on the matrix of regression coefficients, thereby accounting for correlations among response variables. To further improve predictive accuracy and model interpretability, several regularized reduced-rank regression methods have been proposed. However, these existing methods cannot bias the regression coefficients toward the leading principal component directions while accounting for the correlation structure among explanatory variables. In addition, when the explanatory variables exhibit a group structure, the correlation structure within each group cannot be adequately incorporated. To overcome these limitations, we propose a new method that introduces pcLasso into the reduced-rank regression framework. The proposed method improves predictive accuracy by accounting for the correlation among response variables while strongly biasing the matrix of regression coefficients toward principal component directions with large variance. Furthermore, even in settings where the explanatory variables possess a group structure, the proposed method is capable of explicitly incorporating this structure into the estimation process. Finally, we illustrate the effectiveness of the proposed method through numerical simulations and real data application.

翻译：降秩回归通过对回归系数矩阵施加低秩约束来估计回归系数，从而考虑响应变量之间的相关性。为进一步提高预测精度和模型可解释性，已有多种正则化降秩回归方法被提出。然而，这些现有方法在考虑解释变量间相关结构的同时，无法使回归系数偏向于主要的主成分方向。此外，当解释变量呈现群组结构时，每个群组内部的相关结构无法被充分纳入。为克服这些局限性，我们提出一种将pcLasso引入降秩回归框架的新方法。该方法通过考虑响应变量间的相关性，同时使回归系数矩阵强烈偏向具有大方差的主成分方向，从而提高了预测精度。此外，即使在解释变量具有群组结构的情况下，所提方法也能明确地将该结构纳入估计过程。最后，我们通过数值模拟和实际数据应用验证了所提方法的有效性。