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 a mild condition on the sparsity pattern, we compute 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 joins of toric edge ideals.

翻译：因子分析是一种统计技术，通过少量未观测的潜变量来解释观测随机变量之间的相关性。传统全因子分析中，每个观测变量受所有因子影响。然而，许多应用展现出有趣的稀疏模式，即每个观测变量仅依赖部分因子。本文从代数几何视角研究此类稀疏因子分析模型。在稀疏模式的温和条件下，我们计算了对应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，我们识别了可通过2-优雅项序与环面边理想的联合推导出这些关系的格罗布纳基的若干情形。