Rotation to Sparse Loadings using $L^p$ Losses and Related Inference Problems

Exploratory factor analysis (EFA) has been widely used to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that are widely used to find interpretable loading matrices. This paper proposes a new family of oblique rotations based on component-wise $L^p$ loss functions $(0 < p\leq 1)$ that is closely related to an $L^p$ regularised estimator. Model selection and post-selection inference procedures are developed based on the proposed rotation method. When the true loading matrix is sparse, the proposed method tends to outperform traditional rotation and regularised estimation methods in terms of statistical accuracy and computational cost. Since the proposed loss functions are non-smooth, an iteratively reweighted gradient projection algorithm is developed to solve the optimisation problem. Theoretical results are developed that establish the statistical consistency of the estimation, model selection, and post-selection inference. The proposed method is evaluated and compared with regularised estimation and traditional rotation methods via simulation studies. It is further illustrated by an application to the big-five personality assessment.