Modeling the complex relationships between multiple categorical response variables as a function of predictors is a fundamental task in the analysis of categorical data. However, existing methods can be difficult to interpret and may lack flexibility. To address these challenges, we introduce a penalized likelihood method for multivariate categorical response regression that relies on a novel subspace decomposition to parameterize interpretable association structures. Our approach models the relationships between categorical responses by identifying mutual, joint, and conditionally independent associations, which yields a linear problem within a tensor product space. We establish theoretical guarantees for our estimator, including error bounds in high-dimensional settings, and demonstrate the method's interpretability and prediction accuracy through comprehensive simulation studies.

翻译：建模多个分类响应变量与预测变量之间的复杂关系是分类数据分析中的一项基本任务。然而，现有方法往往难以解释且灵活性不足。为应对这些挑战，我们提出了一种用于多元分类响应回归的惩罚似然方法，该方法基于一种新颖的子空间分解来参数化可解释的关联结构。我们的方法通过识别互惠关联、联合关联及条件独立关联来建模分类响应间的关系，从而在张量积空间中形成一个线性问题。我们为所提出的估计量建立了理论保证，包括高维设定下的误差界，并通过全面的模拟研究展示了该方法在可解释性与预测准确性方面的优势。