  Researchers have widely used exploratory factor analysis (EFA) to learn the
latent structure underlying multivariate data. Rotation and regularised
estimation are two classes of methods in EFA that they often use to find
interpretable loading matrices. In this paper we propose 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. We develop model
selection and post-selection inference procedures 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 nonsmooth, we develop an iteratively reweighted gradient
projection algorithm for solving the optimisation problem. We also develop
theoretical results that establish the statistical consistency of the
estimation, model selection, and post-selection inference. We evaluate the
proposed method and compare it with regularised estimation and traditional
rotation methods via simulation studies. We further illustrate it using an
application to the Big Five personality assessment.
