Penalized factor analysis is an efficient technique that produces a factor loading matrix with many zero elements thanks to the introduction of sparsity-inducing penalties within the estimation process. However, sparse solutions and stable model selection procedures are only possible if the employed penalty is non-differentiable, which poses certain theoretical and computational challenges. This article proposes a general penalized likelihood-based estimation approach for single- and multiple-group factor analysis models. The framework builds upon differentiable approximations of non-differentiable penalties, a theoretically founded definition of degrees of freedom, and an algorithm with integrated automatic multiple tuning parameter selection that exploits second-order analytical derivative information. The proposed approach is evaluated in two simulation studies and illustrated using a real data set. All the necessary routines are integrated into the R package penfa.

惩罚因子分析是一种有效的技术，由于在估计过程中引入了稀疏诱导惩罚，它可以生成具有许多零元素的因子加载矩阵。然而，稀疏解和稳定的模型选择程序只有在所采用的惩罚不可微的情况下才有可能，这带来了一定的理论和计算挑战。本文为单组和多组因子分析模型提出了一种基于惩罚似然的通用估计方法。该框架建立在不可微罚的可微近似、自由度的理论上建立的定义以及利用二阶分析导数信息的集成自动多调谐参数选择的算法之上。所提出的方法在两个模拟研究中进行了评估，并使用真实数据集进行了说明。所有必要的例程都集成到R包penfa。