We address the problem of sufficient dimension reduction for feature matrices, which arises often in sensor network localization, brain neuroimaging, and electroencephalography analysis. In general, feature matrices have both row- and column-wise interpretations and contain structural information that can be lost with naive vectorization approaches. To address this, we propose a method called principal support matrix machine (PSMM) for the matrix sufficient dimension reduction. The PSMM converts the sufficient dimension reduction problem into a series of classification problems by dividing the response variables into slices. It effectively utilizes the matrix structure by finding hyperplanes with rank-1 normal matrix that optimally separate the sliced responses. Additionally, we extend our approach to the higher-order tensor case. Our numerical analysis demonstrates that the PSMM outperforms existing methods and has strong interpretability in real data applications.

翻译：本文针对传感器网络定位、脑神经影像学及脑电图分析中常见的特征矩阵充分降维问题展开研究。特征矩阵兼具行向与列向双重解释性，蕴含结构化信息，若采用简单的向量化处理将导致信息丢失。为此，我们提出一种名为主支持矩阵机（PSMM）的矩阵充分降维方法。该方法通过将响应变量划分为切片，将充分降维问题转化为一系列分类问题。PSMM通过寻找秩为1的法向矩阵构造最优分离切片的超平面，从而有效利用矩阵结构。进一步地，我们将该方法推广至高阶张量情形。数值分析表明，PSMM优于现有方法，并在实际数据应用中展现出强可解释性。