Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors. In traditional full factor analysis, each observed variable is influenced by every factor. However, many applications exhibit interesting sparsity patterns, that is, each observed variable only depends on a subset of the factors. In this paper, we study such sparse factor analysis models from an algebro-geometric perspective. Under mild conditions on the sparsity pattern, we examine the dimension of the set of covariance matrices that corresponds to a given model. Moreover, we study algebraic relations among the covariances in sparse two-factor models. In particular, we identify cases in which a Gr\"obner basis for these relations can be derived via a 2-delightful term order and join of toric ideals of graphs.

翻译：因子分析是一种统计技术，它借助较少数量的未观测因子来解释观测随机变量之间的相关性。在传统的完全因子分析中，每个观测变量都受到所有因子的影响。然而，许多应用呈现出有趣的稀疏模式，即每个观测变量仅依赖于因子的一个子集。本文从代数几何的角度研究此类稀疏因子分析模型。在稀疏模式的温和条件下，我们考察了对应于给定模型的协方差矩阵集合的维数。此外，我们研究了稀疏双因子模型中协方差之间的代数关系。特别地，我们识别了能够通过2-愉悦项序和图环面理想的连接来推导这些关系的Gröbner基的情形。