Principal component analysis and factor analysis are fundamental multivariate analysis methods. In this paper a unified framework to connect them is introduced. Under a general latent variable model, we present matrix optimization problems from the viewpoint of loss function minimization, and show that the two methods can be viewed as solutions to the optimization problems with specific loss functions. Specifically, principal component analysis can be derived from a broad class of loss functions including the L2 norm, while factor analysis corresponds to a modified L0 norm problem. Related problems are discussed, including algorithms, penalized maximum likelihood estimation under the latent variable model, and a principal component factor model. These results can lead to new tools of data analysis and research topics.

翻译：主成分分析与因子分析是基础的多变量分析方法。本文提出了连接二者的统一框架。在广义潜变量模型下，我们从损失函数最小化的角度提出了矩阵优化问题，并证明这两种方法可视为特定损失函数优化问题的解。具体而言，主成分分析可从包含L2范数在内的广泛损失函数类别中推导得出，而因子分析则对应于修正的L0范数问题。本文还讨论了相关问题，包括算法、潜变量模型下的惩罚极大似然估计以及主成分因子模型。这些成果可为数据分析提供新工具并开辟新的研究方向。