Irem Portakal: Algebraic sparse factor analysis

Abstract

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 i.e. each observed variable only depends on a subset of the factors. 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 Gröbner basis for these relations can be derived via a 2-delightful term order and joins of toric edge ideals. This is a joint-work with Mathias Drton, Alex Grosdos and Nils Sturma.