Matching Criterion for Identifiability in Sparse Factor Analysis

TL;DR

This paper introduces a new graphical matching criterion for sparse factor analysis models that improves identifiability conditions by leveraging local graph structures, making the process more efficient and applicable.

Contribution

It proposes a novel matching criterion based on bipartite graph structures that enhances identifiability analysis in sparse factor analysis models.

Findings

The matching criterion improves identifiability conditions.

The method is computationally efficient, polynomial in graph size.

It connects sparsity patterns to graphical conditions for factor loadings.

Abstract

Factor analysis models explain dependence among observed variables by a smaller number of unobserved factors. A main challenge in confirmatory factor analysis is determining whether the factor loading matrix is identifiable from the observed covariance matrix. The factor loading matrix captures the linear effects of the factors and, if unrestricted, can only be identified up to an orthogonal transformation of the factors. However, in many applications the factor loadings exhibit an interesting sparsity pattern that may lead to identifiability up to column signs. We study this phenomenon by connecting sparse confirmatory factor analysis models to bipartite graphs and providing sufficient graphical conditions for identifiability of the factor loading matrix up to column signs. In contrast to previous work, our main contribution, the matching criterion, exploits sparsity by operating locally on the graph structure, thereby improving existing conditions. Our criterion is efficiently decidable in time that is polynomial in the size of the graph, when restricting the search steps to sets of bounded size.